research article on weak-bcc-algebrasalgebras. one of very important identities is the identity ( )...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 935097 10 pageshttpdxdoiorg1011552013935097

Research ArticleOn Weak-BCC-Algebras

Janus Thomys1 and Xiaohong Zhang2

1 Institute of Mathematics Wroclaw University of Technology 50-370 Wroclaw Poland2Department of Mathematics College of Art and Sciences Shanghai Maritime Univesrity Shanghai 201306 China

Correspondence should be addressed to Janus Thomys janusthomysgmailcom

Received 3 August 2013 Accepted 27 August 2013

Academic Editors X Guo and Y Lee

Copyright copy 2013 J Thomys and X Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We describe weak-BCC-algebras (also called BZ-algebras) in which the condition (119909lowast119910)lowast119911 = (119909lowast119911)lowast119910 is satisfied only in the casewhen elements 119909 119910 belong to the same branch We also characterize ideals nilradicals and nilpotent elements of such algebras

1 Introduction

BCK-algebras which are a generalization of the notion ofalgebra of sets with the set subtraction as the only fun-damental nonnullary operation and on the other hand thenotion of implication algebra (cf [1]) were defined by Imaiand Iseki in [2] The class of all BCK-algebras does notform a variety To prove this fact Komori introduced in[3] the new class of algebras called BCC-algebras In viewof strong connections with a BIK+-logic BCC-algebras arealso called BIK+-algebras (cf [4] or [5]) Nowadays manymathematicians especially from China Japan and Koreahave been studying various generalizations of BCC-algebrasAll these algebras have one distinguished element and satisfysome common identities playing a crucial role in thesealgebras

One of very important identities is the identity (119909 lowast 119910) lowast

119911 = (119909 lowast 119911) lowast 119910 It holds in BCK-algebras and in somegeneralizations of BCK-algebras but not in BCC-algebrasBCC-algebras satisfying this identity are BCK-algebras (cf[6] or [7]) Therefore it makes sense to consider such BCC-algebras and some of their generalizations for which thisidentity is satisfied only by elements belonging to somesubsets Such study has been initiated by Dudek in [8]

In this paper we will study weak-BCC-algebras in whichthe condition (119909 lowast 119910) lowast 119911 = (119909 lowast 119911) lowast 119910 is satisfied onlyin the case when elements 119909 119910 belong to the same branchWe describe some endomorphisms of such algebras idealsnilradicals and nilpotent elements

2 Basic Definitions and Facts

Definition 1 Aweak-BCC-algebra is a system (119866 lowast 0) of type(2 0) satisfying the following axioms

(i) ((119909 lowast 119910) lowast (119911 lowast 119910)) lowast (119909 lowast 119911) = 0(ii) 119909 lowast 119909 = 0(iii) 119909 lowast 0 = 119909(iv) 119909 lowast 119910 = 119910 lowast 119909 = 0 rArr 119909 = 119910

Weak-BCC-algebras are called BZ-algebras by manymathematicians especially from China and Korea (cf [9] or[10]) but we save the first name because it coincides withthe general concept of names presented in the book [11] foralgebras of logic

A weak-BCC-algebra satisfying the identity

(v) 0 lowast 119909 = 0

is called a BCC-algebra A BCC-algebra with the condition

(vi) (119909 lowast (119909 lowast 119910)) lowast 119910 = 0

is called a BCK-algebraOne can prove (see [6] or [7]) that a BCC-algebra is a

BCK-algebra if and only if it satisfies the identity

(vii) (119909 lowast 119910) lowast 119911 = (119909 lowast 119911) lowast 119910

An algebra (119866 lowast 0) of type (2 0) satisfying the axioms (i)(ii) (iii) (iv) and (vi) is called a BCI-algebra A BCI-algebra

2 The Scientific World Journal

satisfies also (vii) A weak-BCC-algebra is a BCI-algebra ifand only if it satisfies (vii)

Any weak-BCC-algebra can be considered as a partiallyordered set In any weak-BCC-algebra we can define anatural partial order ⩽ putting

119909 ⩽ 119910 lArrrArr 119909 lowast 119910 = 0 (1)

This means that a weak-BCC-algebra can be considered as apartially ordered set with some additional properties

Proposition 2 An algebra (119866 lowast 0) of type (2 0) with a rela-tion ⩽ defined by (1) is a weak-BCC-algebra if and only if forall 119909 119910 119911 isin 119866 the following conditions are satisfied

(i1015840) (119909 lowast 119910) lowast (119911 lowast 119910) ⩽ 119909 lowast 119911(ii1015840) 119909 ⩽ 119909(iii1015840) 119909 lowast 0 = 119909(iv1015840) 119909 ⩽ 119910 and 119910 ⩽ 119909 imply 119909 = 119910

From (i1015840) it follows that in weak-BCC-algebras implica-tions

119909 ⩽ 119910 997904rArr 119909 lowast 119911 ⩽ 119910 lowast 119911 (2)


are satisfied by all 119909 119910 119911 isin 119866A weak-BCC-algebra which is neither BCC-algebra nor

BCI-algebra is called proper Properweak-BCC-algebras haveat least four elements (see [12]) But there are only two weak-BCC-algebras of order four which are not isomorphic

lowast 0 1 2 3

0 0 0 2 2

1 1 0 2 2

2 2 2 0 0

3 3 3 1 0

(4)

lowast 0 1 2 3

0 0 0 2 2

1 1 0 3 3

2 2 2 0 0

3 3 3 1 0

(5)

1

0 2

3

(6)

They are proper because in both cases (3 lowast 2) lowast 1 = (3 lowast

1) lowast 2Since two nonisomorphic weak-BCC-algebras may have

the same partial order they cannot be investigated as algebraswith the operation induced by partial order For exampleweak-BCC-algebras defined by (4) and (5) have the samepartial order but they are not isomorphic

Themethods of construction of weak-BCC-algebras pro-posed in [12] show that for every 119899 ⩾ 4 there exist at least

two proper weak-BCC-algebras of order 119899 which are notisomorphic

The set of all minimal (with respect to ⩽) elements of119866 isdenoted by 119868(119866) Elements belonging to 119868(119866) are called initial

In the investigation of algebras 119866 connected with varioustypes of logics an important role plays the so-called Dudekrsquosmap 120593 119866 rarr 119866 defined by 120593(119909) = 0lowast119909Themain propertiesof this map in the case of weak-BCC-algebras are collected inthe following theorem proved in [13]

Theorem 3 Let 119866 be a weak-BCC-algebra Then

(1) 1205932(119909) ⩽ 119909(2) 119909 ⩽ 119910 rArr 120593(119909) = 120593(119910)(3) 1205933(119909) = 120593(119909)(4) 1205932(119909 lowast 119910) = 1205932(119909) lowast 1205932(119910)(5) 1205932(119909 lowast 119910) = 120593(119910 lowast 119909)(6) 120593(119909) lowast (119910 lowast 119909) = 120593(119910)

for all 119909 119910 isin 119866

Theorem 4 119868(119866) = 119886 isin 119866 1205932(119886) = 119886

Theproof of this theorem is given in [14] Comparing thisresult with Theorem 3(4) we see that 119868(119866) is a subalgebra of119866 that is it is closed under the operationlowast In some situations(see Theorem 21) 119868(119866) is a BCI-algebra

Corollary 5 119868(119866) = 120593(119866) for any weak-BCC-algebra 119866

Proof Indeed if 119909 isin 120593(119866) then 119909 = 120593(119910) for some 119910 isin 119866Thus by Theorem 3 1205932(119909) = 1205933(119910) = 120593(119910) = 119909 Hence1205932(119909) = 119909 that is 119909 isin 119868(119866) So 120593(119866) sub 119868(119866)

Conversely for 119909 isin 119868(119866) we have 119909 = 1205932(119909) = 120593(120593(119909)) =

120593(119910) where 119910 = 120593(119909) isin 119866 Thus 119868(119866) sub 120593(119866) whichcompletes the proof

Thismeans that an element 119886 isin 119866 is an initial element of aweak-BCC-algebra 119866 if and only if it is mentioned in the firstrow (ie in the row corresponding to 0) of the multiplicationtable of 119866

Let 119866 be a weak-BCC-algebra For each 119886 isin 119868(119866) the set

119861 (119886) = 119909 isin 119866 119886 ⩽ 119909 (7)

is called a branch of 119866 initiated by 119886 A branch containingonly one element is called trivial The branch 119861(0) is thegreatest BCC-algebra contained in a weak-BCC-algebra 119866

([8])According to [1 15] we say that a subset 119860 of a BCK-

algebra119866 is an ideal of119866 if (1) 0 isin 119860 (2) 119910 isin 119860 and 119909lowast119910 isin 119860

imply 119909 isin 119860 If 119860 is an ideal then the relation 120579 defined by

119909120579119910 lArrrArr 119909 lowast 119910 119910 lowast 119909 isin 119860 (8)

is a congruence on a BCK-algebra 119866 Unfortunately it is nottrue for weak-BCC-algebras (cf [16]) In connectionwith thisfact Dudek and Zhang introduced in [16] the new concept ofideals These new ideals are called BCC-ideals

The Scientific World Journal 3

Definition 6 Anonempty subset119860 of a weak-BCC-algebra119866is called a BCC-ideal if

(1) 0 isin 119860(2) 119910 isin 119860 and (119909 lowast 119910) lowast 119911 isin 119860 imply 119909 lowast 119911 isin 119860By putting 119911 = 0 we can see that a BCC-ideal is a BCK-

ideal In a BCK-algebra any ideal is a BCC-ideal but in BCC-algebras there are BCC-ideals which are not ideals in theabove sense (cf [16]) It is not difficult to see that 119861(0) is aBCC-ideal of each weak-BCC-algebra

The equivalence classes of a congruence 120579 defined by (8)where 119860 = 119861(0) coincide with branches of 119866 that is 119861(119886) =119862119886for any 119886 isin 119868(119866) (cf [14]) So

119861 (119886) lowast 119861 (119887) = 119909 lowast 119910 119909 isin 119861 (119886) 119910 isin 119861 (119887)

= 119861 (119886 lowast 119887) (9)

In the following part of this paper we will need those twopropositions proved in [14]

Proposition 7 Elements 119909 119910 isin 119866 are in the same branch ifand only if 119909 lowast 119910 isin 119861(0)

Proposition 8 If 119909 119910 isin 119861(119886) then also 119909 lowast (119909 lowast 119910) and 119910 lowast

(119910 lowast 119909) are in 119861(119886)

One of the important classes of weak-BCC-algebras isthe class of the so-called group-like weak-BCC-algebras calledalso antigrouped BZ-algebras [9] that is weak-BCC-algebrascontaining only trivial branches A special case of suchalgebras is group-like BCI-algebras described in [17]

From the results proved in [17] (see also [9]) it followsthat such weak-BCC-algebras are strongly connected withgroups

Theorem 9 An algebra (119866 lowast 0) is a group-like weak-BCC-algebra if and only if (119866 sdot

minus1 0) where 119909 sdot 119910 = 119909 lowast (0 lowast 119910) is agroup Moreover in this case 119909 lowast 119910 = 119909 sdot 119910minus1

Corollary 10 A group (119866 sdotminus1 0) is abelian if and only if thecorresponding weak-BCC-algebra 119866 is a BCI-algebra

Corollary 11 119868(119866) is a maximal group-like subalgebra of eachweak-BCC-algebra 119866

3 Solid Weak-BCC-Algebras

As it is well known in the investigations of BCI-algebras theidentity (vii) plays a very important role It is used in theproofs of almost all theorems but as Dudek noted in hispaper [8] many of these theorems can be proved withoutthis identity Just assume that this identity is fulfilled only byelements belonging to the same branch In this way we obtaina new class of weak-BCC-algebras which are called solid

Definition 12 A weak-BCC-algebra 119866 is called solid if theequation(vii) (119909 lowast 119910) lowast 119911 = (119909 lowast 119911) lowast 119910

is satisfied by all 119909 119910 belonging to the same branch andarbitrary 119911 isin 119866

Any BCI-algebra and any BCK-algebra are solid weak-BCC-algebras A solid weak-BCC-algebra containing onlyone branch is a BCK-algebra To see examples of solid weak-BCC-algebras which are not BCI-algebras one can find themin [8]

Theorem 13 Dudekrsquos map 120593 is an endomorphism of each solidweak-BCC-algebra

Proof Indeed

120593 (119909) lowast 120593 (119910) = (0 lowast 119909) lowast (0 lowast 119910)

= (((119909 lowast 119910) lowast (119909 lowast 119910)) lowast 119909) lowast (0 lowast 119910)

= (((119909 lowast 119910) lowast 119909) lowast (119909 lowast 119910)) lowast (0 lowast 119910)


= ((0 lowast 119910) lowast (119909 lowast 119910)) lowast (0 lowast 119910)


= 0 lowast (119909 lowast 119910) = 120593 (119909 lowast 119910)

(10)

for all 119909 119910 isin 119866

Corollary 14 119868(119866) is a maximal group-like BCI-subalgebra ofeach solid weak BCC-algebra

Proof Comparing Corollaries 5 and 11 we see that 119868(119866) is amaximal group-like subalgebra of each weak BCC-algebra119866Thus by Theorem 9 there exists a group (119868(119866) sdotminus1 0) suchthat 119886 lowast 119887 = 119886 sdot 119887minus1 for 119886 119887 isin 119868(119866) Since 119866 is solid 120593 is itsendomorphism Hence (0 lowast 119886) lowast (0 lowast 119887) = 0 lowast (119886 lowast 119887) for119886 119887 isin 119868(119866) that is 119886minus1 sdot 119887 = (119886 sdot 119887minus1)

minus1

= 119887 sdot 119886minus1 in thecorresponding group The last is possible only in an abeliangroup but in this case (119886 lowast 119887) lowast 119888 = (119886 lowast 119888) lowast 119887 which meansthat 119868(119866) is a BCI-algebra

Definition 15 For 119909 119910 isin 119866 and nonnegative integers 119899 wedefine

1199091199100

= 119909 119909 lowast 119910119899+1

= (119909 lowast 119910119899

) lowast 119910 (11)

Theorem 16 In solid weak-BCC-algebras the following iden-tity

(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = 0 lowast (119909 lowast 119910)119896 (12)

is satisfied for each nonnegative integer 119896

Proof Let119909 isin 119861(119886)Then byTheorem 3 119886 ⩽ 119909 implies 0lowast119909 =

0lowast119886 Suppose that 0lowast119909119896 = 0lowast119886119896 for somenonnegative integer119896 Then also (0 lowast 119886119896) lowast 119909 ⩽ (0 lowast 119886119896) lowast 119886 by (3) Consequently

0 lowast 119909119896+1

= (0 lowast 119909119896

) lowast 119909

= (0 lowast 119886119896

) lowast 119909 ⩽ (0 lowast 119886119896

) lowast 119886 = 0 lowast 119886119896+1

(13)


which means that 0lowast119909119896+1 = 0lowast119886119896+1 because 0lowast119886119896+1 isin 119868(119866)So 0lowast119886119896 = 0lowast119909119896 is valid for all119909 isin 119861(119886) and eachnonnegativeinteger 119896

Similarly 0 lowast 119910119896 = 0 lowast 119887119896 and 0 lowast (119909 lowast 119910)119896

= 0 lowast (119886 lowast 119887)119896

for 119910 isin 119861(119887) and nonnegative integer 119896 Thus a weak-BCC-algebra 119866 satisfies the identity (12) if and only if

(0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 0 lowast (119886 lowast 119887)119896 (14)

holds for 119886 119887 isin 119868(119866) But in view of Corollary 11 andTheorem 9 in the group (119868(119866) sdot

minus1 0) the last equation can

be written in the following form

119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

(15)

Since a weak-BCC-algebra 119866 is solid by Corollary 14 119868(119866)

is a BCI-algebra So the group (119868(119866) sdotminus1 0) is abelian Thusthe above equation is valid for all 119886 119887 isin 119868(119866) Hence (12) isvalid for all 119909 119910 isin 119866 and all nonnegative integers 119896

Corollary 17 Themap 120593119896(119909) = 0lowast119909119896 is an endomorphism of

each solid weak-BCC-algebra

Definition 18 A weak-BCC-algebra for which 120593119896is an endo-

morphism is called 119896-strong In the case 119896 = 1 we say that itis strong

A solid weak-BCC-algebra is strong for every 119896 Theconverse statement is not true

Example 19 The weak-BCC-algebra defined by (4) is notsolid because (3 lowast 2) lowast 1 = (3 lowast 1) lowast 2 but it is strong for every119896 Indeed in this weak-BCC-algebra we have 0 lowast 119909 = 0 for119909 isin 119861(0) 0 lowast 119909 = 2 for 119909 isin 119861(2) and 0 lowast 119909

2 = 0 for all 119909 isin 119866So it is 1-strong and 2-strong Since in this algebra 0lowast119909119896 = 0

for even 119896 and 0 lowast 119909119896 = 0 lowast 119909 for odd 119896 it is strong for every119896

Example 20 Direct computations show that the group-like weak-BCC-algebra induced by the symmetric group 119878

3

(Theorem 9) is 119896-strong for 119896 = 5 and 119896 = 6 but not for119896 = 1 2 3 4 7 8

Theorem 21 A weak-BCC-algebra 119866 is strong if and only if119868(119866) is a BCI-algebra that is if and only if (119868(119866) sdotminus1 0) is anabelian group

Proof Indeed if119866 is strong then (0lowast119886)lowast(0lowast119887) = 0lowast(119886lowast119887)

holds for all 119886 119887 isin 119868(119866) Thus in the group (119868(119866) sdotminus1 0) wehave 119886minus1 sdot 119887 = (119886 sdot 119887minus1)

minus1

= 119887 sdot119886minus1 which means that the group(119868(119866) sdotminus1 0) is abelian Hence

(119886 lowast 119887) lowast 119888 = 119886 sdot 119887minus1

sdot 119888minus1

= 119886 sdot 119888minus1

sdot 119887minus1

= (119886 lowast 119888) lowast 119887

(16)

for all 119886 119887 119888 isin 119868(119866) So (119868(119866) lowast 0) is a BCI-algebraOn the other hand according to Theorem 3 for any 119909 isin

119861(119886) 119910 isin 119861(119887) we have 0 lowast 119909 = 0 lowast 119886 and 0 lowast 119910 = 0 lowast 119887 So

if 119868(119866) is a BCI-algebra then for any 119886 119887 119888 isin 119868(119866) we have(119886 lowast 119887) lowast 119888 = (119886 lowast 119888) lowast 119887 Consequently

(0 lowast 119909) lowast (0 lowast 119910) = (0 lowast 119886) lowast (0 lowast 119887)






= 0 lowast (119886 lowast 119887) = 0 lowast (119909 lowast 119910)

(17)

because 119909 lowast 119910 isin 119861(119886 lowast 119887) This completes the proof

Corollary 22 A strong weak-BCC-algebra is 119896-strong forevery 119896

Proof In a strong weak-BCC-algebra 119866 the group (119868(119866) sdotminus1 0) is abelian and 0 lowast 119911119896 = 0 lowast 119888119896 for every 119911 isin 119861(119888) Thus

(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = (0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

= 0 lowast (119886 lowast 119887)119896

= 0 lowast (119909 lowast 119910)119896

(18)

for all 119909 isin 119861(119886) and 119910 isin 119861(119887)

Example 20 shows that the converse statement is not truethat is there are weak-BCC-algebras which are strong forsome 119896 but not for 119896 = 1

Corollary 23 A weak-BCC-algebra in which 119868(119866) is a BCI-algebra is strong for every 119896

Corollary 24 In any strong weak-BCC-algebra we have

0 lowast (0 lowast 119909119896

) = 0 lowast (0 lowast 119909)119896 (19)

for every 119909 isin 119866 and every natural 119896

4 Ideals of Weak-BCC-Algebras

To avoid repetitions all results formulated in this sectionwill be proved for BCC-ideals Proofs for ideals are almostidentical to proofs for BCC-ideals

Theorem 25 Let 119866 be a weak-BCC-algebra Then 119860 sub 119868(119866)

is an ideal (BCC-ideal) of 119868(119866) if and only if the set theoreticunion of branches 119861(119886) 119886 isin 119860 is an ideal (BCC-ideal) of 119866

Proof Let 119878(119860) denote the set theoretic union of somebranches initiated by elements belonging to 119860 sub 119868(119866) thatis

119878 (119860) = ⋃119886isin119860

119861 (119886) = 119909 isin 119866 119909 isin 119861 (119886) 119886 isin 119860 (20)


By Corollary 11 119868(119866) is a weak-BCC-algebra contained in119866

If 119860 is a BCC-ideal of 119868(119866) then obviously 0 isin 119860 Conse-quently 0 isin 119878(119860) because 0 isin 119861(0) sub 119878(119860) Now let 119910 isin 119878(119860)

and (119909 lowast119910) lowast 119911 isin 119878(119860) for some 119909 119911 isin 119866 Then 119909 isin 119861(119886) 119910 isin

119861(119887) 119911 isin 119861(119888) and (119909lowast119910)lowast119911 isin 119861(119889) for some 119886 119888 isin 119868(119866) and119887 119889 isin 119860Thus (119909lowast119910)lowast119911 isin (119861(119886)lowast119861(119887))lowast119861(119888) = 119861((119886lowast119887)lowast119888)which means that 119861((119886 lowast 119887) lowast 119888) = 119861(119889) since two branchesare equal or disjoint Hence (119886 lowast 119887) lowast 119888 = 119889 isin 119860 so 119886 lowast 119888 isin 119860Therefore 119909 lowast 119911 isin 119861(119886) lowast 119861(119888) = 119861(119886 lowast 119888) sub 119878(119860) This showsthat 119878(119860) is a BCC-ideal of 119866

Conversely let 119878(119860) be a BCC-ideal of119866 If 119886 (119887lowast119886)lowast119888 isin

119860 for some 119886 isin 119860 and 119887 119888 isin 119868(119866) then 119886 isin 119861(119886) sub 119878(119860)(119887 lowast 119886) lowast 119888 isin 119861((119887 lowast 119886) lowast 119888) sub 119878(119860) Hence 119887 lowast 119888 isin 119878(119860) Since119887lowast119888 isin 119868(119866) and 119878(119860)cap119868(119866) = 119860 the above implies 119887lowast119888 isin 119860Thus 119860 is a BCC-ideal of 119868(119866)

119868(119866) is a subalgebra of each weak-BCC-algebra 119866 but itis not an ideal in general

Example 26 It is easy to check that in the weak-BCC-algebra119866 defined by

lowast 0 119886 119887

0 0 0 119887

119886 119886 0 119887

119887 119887 119887 0

(21)

a

b0

(22)

119868(119866) = 0 119887 is not an ideal because 119886 lowast 119887 = 119887 isin 119868(119866) but119886 notin 119868(119866)

The above example suggests the following

Theorem 27 If 119868(119866) is a proper ideal or a proper BCC-idealof a weak-BCC-algebra 119866 then 119866 has at least two nontrivialbranches

Proof Since 0 = 119868(119866) = 119866 at least one branch of 119866 is nottrivial Suppose that only 119861(119887) has more than one elementThen for any 0 = 119886 isin 119868(119866) and 119909 isin 119861(119887) 119909 = 119887 we have119909 lowast 119886 isin 119861(119887) lowast 119861(119886) = 119861(119887 lowast 119886) But by Corollary 11 119868(119866)

is a maximal group-like subalgebra contained in 119866 Thus119887 lowast 119886 isin 119868(119866) and 119887 lowast 119886 = 119887 because in the case 119887 lowast 119886 = 119887 in thecorresponding group (119866 sdot

minus1 0) we obtain 119887 = 119887 sdot 119886 which isimpossible for 119886 = 0Therefore119861(119887lowast119886) = 119861(119887) and119861(119887lowast119886) hasonly one element So 119909lowast119886 = 119887lowast119886 Hence 119909lowast119886 isin 119868(119866) whichaccording to the assumption on 119868(119866) implies 119909 isin 119868(119866) Theobtained contradiction shows that 119868(119866) cannot be an ideal of119866 Consequently it cannot be a BCC-ideal too

Definition 28 A nonempty subset 119860 of a weak-BCC-algebra119866 is called an (119898 119899)-fold 119901-ideal of 119866 if it contains 0 and

(119909 lowast 119911119898

) lowast (119910 lowast 119911119899

) 119910 isin 119860 997904rArr 119909 isin 119860 (23)

An (119899 119899)-fold 119901-ideal is called an 119899-fold 119901-ideal Since(0 0)-fold 119901-ideals coincide with BCK-ideals we will con-sider (119898 119899)-fold119901-ideals only for119898 ⩾ 1 and 119899 ⩾ 1 Moreoverit will be assumed that 119898 = 119899 + 1 because for 119898 = 119899 + 1 wehave (119909lowast119909

119899+1) lowast (0lowast119909119899) = (0lowast119909119899) lowast (0lowast119909119899) = 0 isin 119860 whichimplies 119909 isin 119860 So 119860 = 119866 for every (119899 + 1 119899)-fold 119901-ideal 119860of 119866 Note that the concept of (1 1)-fold 119901-ideals coincideswith the concept of 119901-ideals studied in BCI-algebras (see eg[18] or [19])

Example 29 It is easy to see that in the weak-BCC-algebradefined by (4) the set119860 = 0 1 is an 119899-fold 119901-ideal for every119899 ⩾ 1 It is not an (119898 119899)-fold 119901-ideal where119898 is odd and 119899 iseven because in this case (2 lowast 2

119898) lowast (0 lowast 2119899) isin 119860 and 0 isin 119860but 2 notin 119860

Putting 119911 = 0 in (23) we see that each (119898 119899)-fold 119901-idealof a weak-BCC-algebra is an ideal The converse statement isnot true since as it follows from Theorem 30 proved beloweach (119898 119899)-fold ideal contains the branch 119861(0) which forBCC-ideals is not true

Theorem 30 Any (119898 119899)-fold 119901-ideal contains 119861(0)

Proof Let119860 be an (119898 119899)-fold 119901-ideal of a weak-BCC-algebra119866 Since for every119909 isin 119861(0) from 0 ⩽ 119909 it follows that 0lowast119909 = 0we have

(119909 lowast 119909119898

) lowast (0 lowast 119909119899

) = (0 lowast 119909119898minus1

) lowast (0 lowast 119909119899

) = 0 isin 119860

(24)

which according to (23) gives 119909 isin 119860 Thus 119861(0) sube 119860

Corollary 31 An (119898 119899)-fold 119901-ideal 119860 together with an ele-ment 119909 isin 119860 contains whole branch containing this element

Proof Let 119909 isin 119860 and 119910 be an arbitrary element from thebranch 119861(119886) containing 119909 Then according to Proposition 7we have 119910 lowast 119909 isin 119861(0) sub 119860 Since 119860 is also an ideal the lastimplies 119910 isin 119860 Thus 119861(119886) sub 119860

Corollary 32 For any 119899-fold119901-ideal119860 from 119909 ⩽ 119910 and 119909 isin 119860it follows that 119910 isin 119860

Theorem 33 A nonempty subset 119860 of a solid weak-BCC-algebra 119866 is its (119898 119899)-fold 119901-ideal if and only if

(a) 119868(119860) is an (119898 119899)-fold 119901-ideal of 119868(119866)(b) 119860 = ⋃119861(119886) 119886 isin 119868(119860)

Proof Let 119860 be an (119898 119899)-fold 119901-ideal of 119866 Then clearly119868(119860) = 119860 cap 119868(119866) = 0 is an (119898 119899)-fold 119901-ideal of 119868(119866) ByCorollary 31 119860 is the set theoretic union of all branches 119861(119886)such that 119886 isin 119868(119860) So any (119898 119899)-fold 119901-ideal 119860 satisfies theabove two conditions

Suppose now that a nonempty subset119860 of119866 satisfies thesetwo conditions Let 119909 119910 119911 isin 119866 If 119909 isin 119861(119886) 119910 isin 119861(119887) 119911 isin

119861(119888) and 119910 (119909lowast119911119898)lowast (119910lowast119911119899) isin 119860 then (119909lowast119911119898)lowast (119910lowast119911119899) isin

119861((119886 lowast 119888119898) lowast (119887 lowast 119888119899)) which by (119887) implies 119887 (119886 lowast 119888119898) lowast (119887 lowast

119888119899) isin 119868(119860) This by (119886) gives 119886 isin 119868(119860) So 119861(119886) sub 119860 Hence119909 isin 119860


Note that in some situations the converse of Theorem 30is true

Theorem34 An ideal119860 of a weak-BCC-algebra119866 is its 119899-fold119901-ideal if and only if 119861(0) sub 119860

Proof By Theorem 30 any 119899-fold 119901-ideal contains 119861(0) Onthe other hand if 119860 is an ideal of 119866 and 119861(0) sub 119860 then from119910 isin 119860 and (119909 lowast 119911

119899) lowast (119910 lowast 119911119899) isin 119860 by (i1015840) it follows that

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1

) lowast (119910 lowast 119911119899minus1

)

⩽ (119909 lowast 119911119899minus2


) ⩽ sdot sdot sdot ⩽ 119909 lowast 119910

(25)

so (119909lowast119911119899)lowast(119910lowast119911119899) and 119909lowast119910 as comparable elements are inthe same branch Hence (119909lowast119910)lowast((119909lowast119911119899)lowast(119910lowast119911119899)) isin 119861(0) sub

119860 by Proposition 7 Since (119909 lowast 119911119899) lowast (119910 lowast 119911119899) isin 119860 and 119860 is aBCC-ideal (or a BCK-ideal) (119909lowast119910)lowast((119909lowast119911

119899)lowast(119910lowast119911

119899)) isin 119860

implies 119909 lowast 119910 isin 119860 Consequently 119909 isin 119860 So 119860 is an 119899-fold119901-ideal

Corollary 35 Any ideal containing an 119899-fold 119901-ideal is alsoan 119899-fold 119901-ideal

Proof Suppose that an ideal 119861 contains some 119899-fold 119901-ideal119860 Then 119861(0) sub 119860 sub 119861 which completes the proof

Corollary 36 An ideal 119860 of a weak-BCC-algebra 119866 is its 119899-fold 119901-ideal if and only if the implication

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) isin 119860 997904rArr 119909 lowast 119910 isin 119860 (26)

is valid for all 119909 119910 119911 isin 119866

Proof Let119860be an 119899-fold119901-ideal of119866 Since (119909lowast119911119899)lowast(119910lowast119911119899) ⩽119909 lowast 119910 from (119909 lowast 119911119899) lowast (119910 lowast 119911119899) isin 119860 and by Corollary 32we obtain 119909 lowast 119910 isin 119860 So any 119899-fold 119901-ideal satisfies thisimplication

The converse statement is obvious

Theorem 37 An 119899-fold 119901-ideal is a 119896-fold 119901-ideal for any 119896 ⩽

119899

Proof Similarly as in the previous proof we have

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ sdot sdot sdot ⩽ (119909 lowast 119911119896

) lowast (119910 lowast 119911119896

)

(27)

for every 1 ⩽ 119896 ⩽ 119899 Thus (119909 lowast 119911119899) lowast (119910 lowast 119911119899) and (119909 lowast 119911119896) lowast

(119910 lowast 119911119896) are in the same branch Hence if 119860 is an 119899-fold 119901-ideal and (119909 lowast 119911119896) lowast (119910 lowast 119911119896) isin 119860 then by Corollary 31 also(119909 lowast 119911119899) lowast (119910 lowast 119911119899) isin 119860 This together with 119910 isin 119860 implies119909 isin 119860 Therefore 119860 is a 119896-fold ideal

Theorem 38 119861(0) is the smallest 119899-fold 119901-ideal of each weak-BCC-algebra

Proof Obviously 0 isin 119861(0) If 119910 isin 119861(0) then 0 ⩽ 119910 0 lowast 119911119899 ⩽

119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)

⩽ sdot sdot sdot ⩽ 119909 lowast 0 = 119909

(28)

Thus (119909 lowast 119911119899) lowast (119910 lowast 119911

119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)

means 0 ⩽ (119909lowast119911119899)lowast(119910lowast119911119899) from the above we obtain 0 ⩽ 119909So 119909 isin 119861(0) Hence119861(0) is an 119899-fold119901-ideal ByTheorem 30it is the smallest 119899-fold 119901-ideal of each weak-BCC-algebra

Theorem 39 Let 119866 be a weak-BCC-algebra If 119868(119866) has 119896

elements and 119896 divides |119898 minus 119899| then 119861(0) is an (119898 119899)-fold 119901-ideal of 119866

Proof By Corollary 11 119868(119866) is a group-like subalgebra of 119866Hence if 119868(119866) has 119896 elements then in the group (119868(119866) sdotminus1 0)

connected with 119868(119866) (Theorem 9) we have 119887119896119904 = 0 for every119887 isin 119868(119866) and any integer 119904

At first we consider the case119898 ⩾ 119899 If (119909lowast119911119898)lowast(119910lowast119911119899) isin

119861(0) for some 119909 isin 119861(119886) 119910 isin 119861(0) 119911 isin 119861(119888) then by (i1015840) wehave (119909lowast119911119898)lowast(119910lowast119911119899) ⩽ (119909lowast119911119898minus119899)lowast119910 Hence (119909lowast119911119898minus119899)lowast119910

and (119909 lowast 119911119898) lowast (119910 lowast 119911119899) as comparable elements are in thesame branch Consequently ((119909 lowast 119911119898minus119899) lowast 119910) lowast ((119909 lowast 119911119898) lowast

(119910lowast119911119899)) isin 119861(0) (Proposition 7) Since119861(0) is an ideal in eachweak-BCC-algebra from the last we obtain (119909 lowast 119911119898minus119899) lowast 119910 isin

119861(0) and consequently 119909 lowast 119911119898minus119899 isin 119861(0) But 119909 lowast 119911119898minus119899 isin

119861(119886 lowast 119888119898minus119899) so 119861(0) = 119861(119886 lowast 119888119898minus119899) that is 0 = 119886 lowast 119888119898minus119899This in the group (119868(119866) sdotminus1 0) connected with 119868(119866) gives 0 =

119886 sdot 119888119899minus119898 = 119886 So 119909 isin 119861(0)Now let119898 lt 119899 Then (119909 lowast 119911119898) lowast (119910 lowast 119911119899) ⩽ 119909lowast (119910lowast 119911119899minus119898)

This similarly as in the previous case for (119909lowast119911119898)lowast(119910lowast119911119899) isin

119861(0) gives (119909 lowast (119910 lowast 119911119899minus119898)) lowast ((119909 lowast 119911119898) lowast (119910 lowast 119911119899)) isin 119861(0)Consequently 119909 lowast (119910 lowast 119911119899minus119898) isin 119861(0) cap 119861(119886 lowast (0 lowast 119888119899minus119898)) So0 = 119886 lowast (0 lowast 119888119899minus119898) This in the group (119868(119866) sdotminus1 0) implies0 = 119886 sdot 119888119899minus119898 = 119886 Hence 119909 isin 119861(0)

The proof is complete

The assumption on the number of elements of the set 119868(119866)

is essential if 119896 is not a divisor of |119898 minus 119899| then 119861(0) may notbe an (119898 119899)-fold 119901-ideal

Example 40 The solid weak-BCC-algebra 119866 defined by

lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)


is proper because (3lowast1)lowast4 = (3lowast4)lowast1The set 119868(119866) has threeelements The set 119861(0) = 0 2 is an 119899-fold 119901-ideal for everynatural 119899 but it is not a (3 2)-fold ideal because (1 lowast 1

3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)

In the case when 119861(0) has only one element the equiva-lence relation induced by 119861(0) has one-element equivalenceclasses Since these equivalence classes are branches a weak-BCC-algebra with this property is group-like Direct compu-tations show that in this case 119861(0) is an 119899-fold 119901-ideal forevery natural 119899

This observation together with the just proved resultssuggests simple characterization of group-like weak-BCC-algebras

Theorem 41 A weak-BCC-algebra 119866 is group-like if and onlyif for some 119899 ⩾ 1 and all 119909 119911 isin 119866

(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)

Proof Let 119866 be a weak-group-like BCC-algebra Then 119866 =

119868(119866) which means that 119866 has a discrete order that is 119909 ⩽ 119910

implies119909 = 119910 Since for119909 119910 119911 isin 119866wehave (119909lowast119911119899)lowast(119910lowast119911119899) ⩽119909 lowast 119910 a group-like weak-BCC-algebra satisfies the identity(119909 lowast 119911

119899) lowast (119910 lowast 119911119899) = 119909 lowast 119910 In particular for 119910 = 0 we have(119909 lowast 119911119899) lowast (0 lowast 119911119899) = 119909 lowast 0 = 119909 So (119909 lowast 119911119899) lowast (0 lowast 119911119899) = 0

implies 119909 = 0Conversely if the above implication is valid for all 119909 119911 isin

119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)

gives 0 ⩽ 119909 This according to the assumption implies 119909 = 0Hence 119861(0) = 0 which means that 119866 is group-like

Remember that an ideal 119860 of a weak-BCC-algebra iscalled closed if 0 lowast 119909 isin 119860 for every 119909 isin 119860 that is if 120593(119860) sub 119860

Theorem 42 For an (119898 119899)-fold 119901-ideal 119860 of a solid weak-BCC-algebra 119866 the following statements are equivalent

(1) 119860 is a closed (119898 119899)-fold 119901-ideal of 119866(2) 119868(119860) is a closed (119898 119899)-fold 119901-ideal of 119868(119866)(3) 119868(119860) is a subalgebra of 119868(119866)(4) 119860 is a subalgebra of 119866

Proof The implication (1) rArr (2) follows fromTheorem 33(2) rArr (3) Observe first that 119868(119860) is a closed BCK-ideal

of 119868(119866) and 119886 lowast 119887 = 119888 isin 119868(119866) for any 119886 119887 isin 119868(119860) Since 119868(119866)

is a group-like subalgebra of 119866 (Corollary 11) in the group(119868(119866) sdot

minus1 0) we have 119888 = 119886 sdot 119887minus1 (Theorem 9) which meansthat 119888 sdot 119887 = 119886 isin 119868(119860) Thus

119888 sdot (0 sdot 119887) = 119888 lowast (0 sdot 119887)minus1

= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)

Hence 119888lowast (0lowast119887) isin 119868(119860) But 0lowast119887 isin 119868(119860) and 119868(119860) is a BCK-ideal of 119868(119866) therefore 119888 isin 119868(119860) Consequently 119886 lowast 119887 isin 119868(119860)

for every 119886 119887 isin 119868(119860) So 119868(119860) is a subalgebra of 119868(119866)

(3) rArr (4) 119868(119860) sub 119860 so 0 isin 119860 Let 119909 isin 119861(119886) 119910 isin 119861(119887) If119909 119910 isin 119860 then 119886 119887 isin 119868(119860) and by the assumption 119886lowast119887 isin 119868(119860)From this we obtain 119909 lowast 119910 isin 119861(119886) lowast 119861(119887) = 119861(119886 lowast 119887) whichtogether with Theorem 33 proves 119909 lowast 119910 isin 119860 Hence 119860 is asubalgebra of 119866

The implication (4) rArr (1) is obvious

5 Nilpotent Weak-BCC-Algebras

A special role in weak-BCC-algebras play elements having afinite ldquoorderrdquo that is elements for which there exists somenatural 119896 such that 0 lowast 119909119896 = 0 We characterize sets of suchelements and prove that the properties of such elements canbe described by the properties of initial elements of branchescontaining these elements

Definition 43 An element 119909 of a weak-BCC-algebra 119866 iscalled nilpotent if there exists some positive integer 119896 suchthat 0 lowast 119909

119896 = 0 The smallest 119896 with this property is called thenilpotency index of 119909 and is denoted by 119899(119909) A weak-BCC-algebra in which all elements are nilpotent is called nilpotent

By119873119896(119866) we denote the set of all nilpotent elements 119909 isin

119866 such that 119899(119909) = 119896 119873(119866) denotes the set of all nilpotentelements of 119866 It is clear that119873

1(119866) = 119861(0)

Example 44 In the weak-BCC-algebras defined by (4) and(5) we have 119899(0) = 119899(1) = 1 119899(2) = 119899(3) = 2

Example 45 In the weak-BCC-algebra defined by

lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)

there are no elements with 119899(119909) = 2 but there are threeelements with 119899(119909) = 3 and three with 119899(119909) = 1

Proposition 46 Elements belonging to the same branch havethe same nilpotency index

Proof Let 119909 isin 119861(119886) Then 119886 ⩽ 119909 which by Theorem 3implies 0 lowast 119886 = 0 lowast 119909 This together with 119886 ⩽ 119909 gives0lowast1199092 ⩽ (0lowast119886)lowast119909 ⩽ 0lowast1198862 Hence 0lowast1199092 ⩽ 0lowast1198862 In the samemanner from 0lowast119909119896 ⩽ 0lowast119886119896 it follows that 0lowast119909119896+1 ⩽ 0lowast119886119896+1which by induction proves 0lowast119909119898 ⩽ 0lowast119886119898 for every 119909 isin 119861(119886)

and any natural 119898 Thus 0 lowast 119886119898 = 0 implies 0 lowast 119909119898 = 0 Onthe other hand from 0 lowast 119909119898 = 0 we obtain 0 ⩽ 0 lowast 119886119898 Thisimplies 0 = 0 lowast 119886119898 since 0 0 lowast 119886119898 isin 119868(119866) and elements of119868(119866) are incomparable Therefore 0 lowast 119909119898 = 0 if and only if0 lowast 119886119898 = 0 So 119899(119909) = 119899(119886) for every 119909 isin 119861(119886)


Corollary 47 A weak-BCC-algebra 119866 is nilpotent if and onlyif its subalgebra 119868(119866) is nilpotent

Corollary 48 119909 isin 119861(119886) cap 119873119896(119866) rArr 119861(119886) sub 119873

119896(119866)

The above results show that the study of nilpotency ofa given weak-BCC-algebras can be reduced to the study ofnilpotency of its initial elements

Proposition 49 Let 119866 be a weak-BCC-algebra If 119868(119866) is aBCI-algebra then119873

119896(119866) is a subalgebra and a BCC-ideal of 119866

for every 119896

Proof Obviously 0 isin 119873119896(119866) for every 119896 Let 119909 119910 isin 119873

119896(119866)

Then 0 lowast 119909119896= 0 lowast 119910

119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=

0 lowast 119887119896 = 0 for some 119886 119887 isin 119868(119866) Since 119868(119866) is a BCI-algebrabyTheorem 16 we have 0 = (0 lowast 119886119896) lowast (0 lowast 119887119896) = 0 lowast (119886 lowast 119887)

119896Hence 119886 lowast 119887 isin 119873

119896(119866) Consequently 119909 lowast 119910 isin 119861(119886) lowast 119861(119887) =

119861(119886 lowast 119887) sub 119873119896(119866) So119873

119896(119866) is a subalgebra of 119866

Now let 119909 isin 119861(119886) 119910 isin 119861(119887) 119911 isin 119861(119888) If 119910 (119909 lowast 119910) lowast 119911 isin

119873119896(119866) then also 119887 (119886 lowast 119887) lowast 119888 isin 119873

119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)

= 0 lowast ((119886 lowast 119887) lowast 119888)119896

= 0

(36)

which implies 119886lowast 119888 isin 119873119896(119866) This together with Corollary 48

implies 119909lowast119911 isin 119861(119886lowast 119888) sub 119873119896(119866) Therefore119873

119896(119866) is a BCC-

ideal of 119866 Clearly it is a BCK-ideal too

Corollary 50 119873119896(119866) is a subalgebra of each solid weak-BCC-

algebra

Proposition 51 119873(119866) is a subalgebra of each weak-BCC-algebra 119866 in which 119868(119866) is a BCI-algebra

Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896

the set 119873(119866) is nonempty Let 119909 isin 119861(119886) 119910 isin 119861(119887) If 119909 119910 isin

119873(119866) and 119899(119909) = 119898 119899(119910) = 119899 then 0 lowast 119909119898 = 0 lowast 119910119899 = 0

From this by Proposition 46 we obtain 0 lowast 119886119898

= 0 lowast 119887119899= 0

which in the group (119868(119866) sdotminus1 0) can be written in the form119886minus119898 = 119887minus119899 = 0 But 119868(119866) is a BCI-algebra hence (119868(119866) sdotminus1 0)

is an abelian group Thus

0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)

by Theorem 9 Hence 119886 lowast 119887 isin 119873(119866) This implies 119909 lowast 119910 isin

119861(119886 lowast 119887) sub 119873(119866) Therefore119873(119866) is a subalgebra of 119866

Corollary 52 119873(119866) is a subalgebra of each solid weak-BCC-algebra

Corollary 53 Any solid weak-BCC-algebra119866with finite 119868(119866)

is nilpotent

Proof Indeed 119868(119866) is a maximal group-like BCI-algebracontained in any solid weak-BCC-algebra Hence the group

(119868(119866) sdotminus1 0) is abelian If it is finite then each of its element

has finite order 119896Thus 0lowast119886119896 = 0sdot119886minus119896 = 0 for every 119886 isin 119868(119866)Consequently 119861(119886) sub 119873

119896(119866) sub 119873(119866) for every 119886 isin 119868(119866)

Therefore 119866 = 119873(119866)

Corollary 54 A solid weak-BCC-algebra119866 is nilpotent if andonly if each element of the group (119868(119866) sdotminus1 0) has finite order

Corollary 55 In a solid weak-BCC-algebra 119866 the nilpotencyindex of each 119909 isin 119873(119866) is a divisor of 119862119886119903119889(119868(119866))

6 119896-Nilradicals of Solid Weak-BCC-Algebras

The theory of radicals in BCI-algebras was considered bymanymathematicians from China (cf [18]) Obtained resultsshow that this theory is almost parallel to the theory ofradicals in rings But results proved for radicals in BCI-algebras cannot be transferred to weak-BCC-algebras

In this section we characterize one analog of nilradicalsin weak-BCC-algebras Further this characterization will beused to describe some ideals of solid weak-BCC-algebras

We begin with the following definition

Definition 56 Let119860 be a subset of solidweak-BCC-algebra119866For any positive integer 119896 by a 119896-nilradical of 119860 denoted by[119860 119896] we mean the set of all elements of 119866 such that 0 lowast 119909

119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)

Example 57 In the weak-BCC-algebra 119866 defined inExample 44 for119860 = 0 119886 and any natural 119896 we have [119860 3119896+

1] = [119860 3119896+2] = 119861(0) [119860 3119896] = 119866 But for 119861 = 119886 119890 we get[119861 3119896 + 1] = 119889 [119861 3119896 + 2] = 119861(119888) The set [119861 3119896] is empty


lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)

is proper because (3lowast4)lowast5 = (3lowast5)lowast4 In this algebra each119896-nilradical of 119860 = 0 5 is equal to 119866 each 119896-nilradical of119861 = 1 4 is empty

The first question is when for a given nonempty set 119860its 119896-nilradical is also nonempty The answer is given in thefollowing proposition


Proposition 59 A 119896-nilradical [119860 119896] of a nonempty subset 119860of a weak-BCC-algebra119866 is nonempty if and only if119860 containsat least one element 119886 isin 119868(119866)

Proof From the proof of Theorem 16 it follows that 0 lowast 119909119896 =

0 lowast 119886119896 for every 119909 isin 119861(119886) and any positive 119896 So 119909 isin [119860 119896] ifand only if 0 lowast 119886119896 isin 119860 The last means that 0 lowast 119886119896 isin 119860 cap 119868(119866)

because 119868(119866) is a subalgebra of 119866

Corollary 60 [119868(119866) 119896] = 119866 for every 119896

Proof Indeed 0 lowast 119909119896 = 0 lowast 119886119896 isin 119868(119866) for every 119909 isin 119866 Thus119909 isin [119868(119866) 119896]

Corollary 61 If 119868(119866) has 119899 elements then [119860 119899] = 119866 for anysubset 119860 of 119866 containing 0 and [119860 119899] = 0 if 0 notin 119860

Proof Similarly as in previous proofs we have 0lowast119909119896 = 0lowast119886119896

for every 119909 isin 119861(119886) and any 119896 Since 0 lowast 119886119896 isin 119868(119866) and119868(119866) is a group-like subalgebra of 119866 0 lowast 119909119896 = 119886minus119896 in thegroup (119868(119866) sdotminus1 0) (Theorem 9) If 119868(119866) has 119899 elements thenobviously 0 lowast 119909119899 = 119886minus119899 = 0 isin 119860 Hence 119909 isin [119860 119899] Thiscompletes the proof

Corollary 62 Let 119909 isin 119861(119886) Then 119909 isin [119860 119896] if and only if119861(119886) sub [119860 119896]

Proof Since 0 lowast 119909119896 = 0 lowast 119886119896 we have 119909 isin [119860 119896] hArr 119886 isin

[119860 119896]

Corollary 63 [119860 119896] = ⋃119861(119886) 0 lowast 119886119896 isin 119860

Proposition 64 Let119866 be a solid weak-BCC-algebraThen forevery positive integer 119896 and any subalgebra119860 of119866 a 119896-nilradical[119860 119896] is a subalgebra of 119866 such that 119860 sube [119860 119896]

Proof Let 119909 119910 isin [119860 119896] Then 0 lowast 119909119896 0 lowast 119910119896 isin 119860 and 0 lowast

(119909 lowast 119910)119896

= (0 lowast 119909119896) lowast (0 lowast 119910119896) isin 119860 by Theorem 16 Hence119909 lowast 119910 isin [119860 119896] Clearly 119860 sube [119860 119896]

Proposition 65 In a solid weak-BCC-algebra a 119896-nilradicalof an ideal is also an ideal

Proof Let 119860 be a BCC-ideal of 119866 If 119910 isin [119860 119896] and (119909 lowast 119910) lowast

119911 isin [119860 119896] then 0 lowast 119910119896 isin 119860 and 119860 ni 0 lowast ((119909 lowast 119910) lowast 119911)119896

=

((0 lowast 119909119896) lowast (0 lowast 119910119896)) lowast (0 lowast 119911119896) by Theorem 16 Hence 119860 ni

(0 lowast 119909119896) lowast (0 lowast 119911119896) = 0 lowast (119909 lowast 119911)119896 Thus 119909 lowast 119911 isin [119860 119896]

Note that the last two propositions are not true for weak-BCC-algebras which are not solid

Example 66 The weak-BCC-algebra 119866 induced by the sym-metric group 119878

3is not solid because 119878

3is not an abelian group

(Corollary 14) Routine calculations show that 119860 = 0 3 is asubalgebra and a BCC-ideal of this weak-BCC-algebra but[119860 3] = 0 1 2 3 is neither ideal nor subalgebra

Theorem 67 In a solid weak-BCC-algebra a 119896-nilradical ofan (119898 119899)-fold 119901-ideal is also an (119898 119899)-fold 119901-ideal

Proof By Proposition 65 a 119896-nilradical of an (119898 119899)-fold 119901-ideal 119860 of 119866 is an ideal of 119866 If 119910 (119909 lowast 119911

119898) lowast (119910 lowast 119911119899) isin [119860 119896]then 0 lowast 119910119896 0 lowast ((119909 lowast 119911119898) lowast (119910 lowast 119911119899)

119896

) isin 119860 Hence applyingTheorem 16 we obtain

((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)

Thus 0 lowast 119909119896 isin 119860 So 119909 isin [119860 119896]

Note that in general a 119896-nilradical [119860 119896] of an ideal 119860does not save all properties of an ideal 119860 For example if anideal 119860 is a horizontal ideal that is 119909 isin 119860 cap 119861(0) hArr 119909 = 0then a 119896-nilradical [119860 119896]may not be a horizontal ideal Suchsituation takes place in a weak-BCC-algebra defined by (34)In this algebra we have 0 lowast 119909

3 = 0 for all elements Hence119909 isin [119860 3] cap 119861(0) means that 0 lowast 1199093 isin 119860 and 119909 isin 119861(0) whichis also true for 119909 = 0

Nevertheless properties of many main types of ideals aresaved by their 119896-nilradicals Below we present the list of themain types of ideals considered in BCI-algebras and weak-BCC-algebras

Definition 68 An ideal 119860 of a weak-BCC-algebra 119866 is called

(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if

(119909 lowast 119910) lowast 119911 119910 lowast 119911 isin 119860 997904rArr 119909 isin 119860 (43)

(iii) quasiassociative if

119909 lowast (119910 lowast 119911) 119910 isin 119860 997904rArr 119909 lowast 119911 isin 119860 (44)

(iv) closed if

119909 isin 119860 997904rArr 0 lowast 119909 isin 119860 (45)

(v) commutative if

119909 lowast 119910 isin 119860 997904rArr 119909 lowast (119910 lowast (119910 lowast 119909)) isin 119860 (46)

(vi) subcommutative if

119910 lowast (119910 lowast (119909 lowast (119909 lowast 119910))) =isin 119860 997904rArr 119909 lowast (119909 lowast 119910) isin 119860 (47)

(vii) implicative if

(119909 lowast 119910) lowast 119911 119910 lowast 119911 isin 119860 997904rArr 119909 lowast 119911 isin 119860 (48)

(viii) subimplicative if

(119909 lowast (119909 lowast 119910)) lowast (119910 lowast 119909) isin 119860 997904rArr 119910 lowast (119910 lowast 119909) isin 119860 (49)


(ix) weakly implicative if

(119909 lowast (119910 lowast 119909)) lowast (0 lowast (119910 lowast 119909)) isin 119860 997904rArr 119909 isin 119860 (50)

(x) obstinate if

119909 119910 notin 119860 997904rArr 119909 lowast 119910 119910 lowast 119909 isin 119860 (51)

(xi) regular if

119909 lowast 119910 119909 isin 119860 997904rArr 119910 isin 119860 (52)

(xii) strong if

119909 isin 119860 119910 isin 119883 minus 119860 997904rArr 119909 lowast 119910 isin 119883 minus 119860 (53)

for all 119909 119910 119911 isin 119866

Definition 69 We say that an ideal 119860 of a weak-BCC-algebra119866 has the property P if it is one of the above types thatis if it satisfies one of implications mentioned in the abovedefinition

Theorem 70 If an ideal 119860 of a solid weak-BCC-algebra 119866

has the property P then its 119896-nilradical [119860 119896] also has thisproperty

Proof (1) 119860 is antigrouped Let 1205932(119909) isin [119860 119896] Then 0 lowast

(1205932(119909))119896

isin 119860 Since by Theorem 3 1205932 is an endomorphismof each weak-BCC-algebra we have

1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)

Thus 1205932(0 lowast 119909119896) isin 119860 which according to the definitionimplies 0 lowast 119909119896 isin 119860 Hence 119909 isin [119860 119896]

(2) 119860 is associative If (119909 lowast 119910) lowast 119911 119910 lowast 119911 isin [119860 119896] then0 lowast ((119909 lowast 119910) lowast 119911)

119896

isin 119860 and 0 lowast (119910 lowast 119911)119896

isin 119860 which in view ofTheorem 16 means that ((0 lowast 119909119896) lowast (0 lowast 119910119896)) lowast (0 lowast 119911119896) isin 119860

and (0 lowast 119910119896) lowast (0 lowast 119911119896) isin 119860 Since an ideal 119860 is associativethis implies 0 lowast 119909119896 isin 119860 that is 119909 isin [119860 119896]

(3) 119860 is quasiassociative Similarly as in the previous case119909 lowast (119910 lowast 119911) 119910 isin [119860 119896]means that 0 lowast (119909 lowast (119910 lowast 119911))

119896

isin 119860 and0 lowast 119910119896 isin 119860 Hence (0 lowast 119909119896) lowast ((0 lowast 119910119896) lowast (0 lowast 119911119896)) isin 119860 Thisimplies 0 lowast (119909 lowast 119911)

119896

= (0 lowast 119909119896) lowast (0 lowast 119911119896) isin 119860 Consequently119909 lowast 119911 isin [119860 119896]

(4) 119860 is closed Let 119909 isin [119860 119896] Then 0 lowast 119909119896 isin 119860 Thus

0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)

So 0 lowast 119909 isin [119860 119896](5) 119860 is commutative Let 119909 lowast 119910 isin [119860 119896] Then 0 lowast (119909 lowast

119910)119896

isin 119860 From this we obtain (0 lowast 119909119896) lowast (0 lowast 119910119896) isin 119860 whichgives 0 lowast (119909 lowast (119910 lowast (119910 lowast 119909)))

119896

= (0 lowast 119909119896) lowast ((0 lowast 119910119896) lowast ((0 lowast

119910119896) lowast (0 lowast 119909119896))) isin 119860 Hence 119909 lowast (119910 lowast (119910 lowast 119909)) isin [119860 119896]For other types of ideals the proof is very similar

References

[1] K Iseki and S Tanaka ldquoAn introduction to the theory of BCK-algebrasrdquoMathematica Japonica vol 23 pp 1ndash26 1978

[2] Y Imai and K Iseki ldquoOn axiom system of propositional calculirdquoProceedings of the Japan Academy vol 42 pp 19ndash22 1966

[3] Y Komori ldquoThe class of BCC-algebras is not varietyrdquo Mathe-matica Japonica vol 29 pp 391ndash394 1984

[4] X H Zhang ldquoBIK+-logic and non-commutative fuzzy logicsrdquoFuzzy Systems Math vol 21 pp 31ndash36 2007

[5] X H Zhang and W A Dudek ldquoFuzzy BIK+-logic and non-commutative fuzzy logicsrdquo Fuzzy Systems Math vol 23 pp 9ndash20 2009

[6] W A Dudek ldquoOn BCC-algebrasrdquo Logique et Analyse vol 129-130 pp 103ndash111 1990

[7] WADudek ldquoOn proper BCC-algebrasrdquoBulletin of the Instituteof Mathematics Academia Sinica vol 20 pp 137ndash150 1992

[8] W A Dudek ldquoSolid weak BCC-algebrasrdquo International Journalof Computer Mathematics vol 88 no 14 pp 2915ndash2925 2011

[9] X H Zhang and R Ye ldquoBZ-algebras and groupsrdquo Journal ofMathematical and Physical Sciences vol 29 pp 223ndash233 1995

[10] X H Zhang Y Q Wang and B Y Shen ldquoBZ-algebras of order5rdquo East China University of Science and Technology vol 29 pp50ndash53 2003

[11] A Iorgulescu Algebras of Logic as BCK Algebras Academy ofEconomic Studies Bucharest Romania 2008

[12] WADudek ldquoRemarks on the axioms system for BCI-algebrasrdquoPrace Naukowe WSP w Częstochowie Matematyka vol 2 pp46ndash61 1996

[13] W A Dudek X Zhang and Y G Wang ldquoIdeals and atoms ofBZ-algebrasrdquoMathematica Slovaca vol 59 no 4 pp 387ndash4042009

[14] W A Dudek B Karamdin S A Bhatti and K P ShumldquoBranches and ideals of weak BCC-algebrasrdquo Algebra Collo-quium vol 18 no 1 pp 899ndash914 2011

[15] K Iseki and S Tanaka ldquoIdeal theory of BCK-algebrasrdquo Mathe-matica Japonica vol 21 pp 351ndash366 1976

[16] W A Dudek and X H Zhang ldquoOn ideals and congruences inBCC-algebrasrdquo Czechoslovak Mathematical Journal vol 48 no1 pp 21ndash29 1998

[17] W A Dudek ldquoOn group-like BCI-algebrasrdquo DemonstratioMathematica vol 21 pp 369ndash376 1988

[18] Y S Huang BCI-Algebra Science Press Beijing China 2006[19] X H Zhang J Hao and S A Bhatti ldquoOn p-ideals of a BCI-

algebrardquo Punjab University Journal of Mathematics vol 27 pp121ndash128 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


satisfies also (vii) A weak-BCC-algebra is a BCI-algebra ifand only if it satisfies (vii)

Any weak-BCC-algebra can be considered as a partiallyordered set In any weak-BCC-algebra we can define anatural partial order ⩽ putting

119909 ⩽ 119910 lArrrArr 119909 lowast 119910 = 0 (1)

This means that a weak-BCC-algebra can be considered as apartially ordered set with some additional properties

Proposition 2 An algebra (119866 lowast 0) of type (2 0) with a rela-tion ⩽ defined by (1) is a weak-BCC-algebra if and only if forall 119909 119910 119911 isin 119866 the following conditions are satisfied

(i1015840) (119909 lowast 119910) lowast (119911 lowast 119910) ⩽ 119909 lowast 119911(ii1015840) 119909 ⩽ 119909(iii1015840) 119909 lowast 0 = 119909(iv1015840) 119909 ⩽ 119910 and 119910 ⩽ 119909 imply 119909 = 119910

From (i1015840) it follows that in weak-BCC-algebras implica-tions



are satisfied by all 119909 119910 119911 isin 119866A weak-BCC-algebra which is neither BCC-algebra nor

BCI-algebra is called proper Properweak-BCC-algebras haveat least four elements (see [12]) But there are only two weak-BCC-algebras of order four which are not isomorphic

lowast 0 1 2 3

0 0 0 2 2

1 1 0 2 2

2 2 2 0 0

3 3 3 1 0

(4)

lowast 0 1 2 3

0 0 0 2 2

1 1 0 3 3

2 2 2 0 0

3 3 3 1 0

(5)

1

0 2

3

(6)

They are proper because in both cases (3 lowast 2) lowast 1 = (3 lowast

1) lowast 2Since two nonisomorphic weak-BCC-algebras may have

the same partial order they cannot be investigated as algebraswith the operation induced by partial order For exampleweak-BCC-algebras defined by (4) and (5) have the samepartial order but they are not isomorphic

Themethods of construction of weak-BCC-algebras pro-posed in [12] show that for every 119899 ⩾ 4 there exist at least

two proper weak-BCC-algebras of order 119899 which are notisomorphic

The set of all minimal (with respect to ⩽) elements of119866 isdenoted by 119868(119866) Elements belonging to 119868(119866) are called initial

In the investigation of algebras 119866 connected with varioustypes of logics an important role plays the so-called Dudekrsquosmap 120593 119866 rarr 119866 defined by 120593(119909) = 0lowast119909Themain propertiesof this map in the case of weak-BCC-algebras are collected inthe following theorem proved in [13]

Theorem 3 Let 119866 be a weak-BCC-algebra Then

(1) 1205932(119909) ⩽ 119909(2) 119909 ⩽ 119910 rArr 120593(119909) = 120593(119910)(3) 1205933(119909) = 120593(119909)(4) 1205932(119909 lowast 119910) = 1205932(119909) lowast 1205932(119910)(5) 1205932(119909 lowast 119910) = 120593(119910 lowast 119909)(6) 120593(119909) lowast (119910 lowast 119909) = 120593(119910)

for all 119909 119910 isin 119866

Theorem 4 119868(119866) = 119886 isin 119866 1205932(119886) = 119886

Theproof of this theorem is given in [14] Comparing thisresult with Theorem 3(4) we see that 119868(119866) is a subalgebra of119866 that is it is closed under the operationlowast In some situations(see Theorem 21) 119868(119866) is a BCI-algebra

Corollary 5 119868(119866) = 120593(119866) for any weak-BCC-algebra 119866

Proof Indeed if 119909 isin 120593(119866) then 119909 = 120593(119910) for some 119910 isin 119866Thus by Theorem 3 1205932(119909) = 1205933(119910) = 120593(119910) = 119909 Hence1205932(119909) = 119909 that is 119909 isin 119868(119866) So 120593(119866) sub 119868(119866)

Conversely for 119909 isin 119868(119866) we have 119909 = 1205932(119909) = 120593(120593(119909)) =

120593(119910) where 119910 = 120593(119909) isin 119866 Thus 119868(119866) sub 120593(119866) whichcompletes the proof

Thismeans that an element 119886 isin 119866 is an initial element of aweak-BCC-algebra 119866 if and only if it is mentioned in the firstrow (ie in the row corresponding to 0) of the multiplicationtable of 119866

Let 119866 be a weak-BCC-algebra For each 119886 isin 119868(119866) the set

119861 (119886) = 119909 isin 119866 119886 ⩽ 119909 (7)

is called a branch of 119866 initiated by 119886 A branch containingonly one element is called trivial The branch 119861(0) is thegreatest BCC-algebra contained in a weak-BCC-algebra 119866

([8])According to [1 15] we say that a subset 119860 of a BCK-

algebra119866 is an ideal of119866 if (1) 0 isin 119860 (2) 119910 isin 119860 and 119909lowast119910 isin 119860

imply 119909 isin 119860 If 119860 is an ideal then the relation 120579 defined by

119909120579119910 lArrrArr 119909 lowast 119910 119910 lowast 119909 isin 119860 (8)

is a congruence on a BCK-algebra 119866 Unfortunately it is nottrue for weak-BCC-algebras (cf [16]) In connectionwith thisfact Dudek and Zhang introduced in [16] the new concept ofideals These new ideals are called BCC-ideals







= 119861 (119886 lowast 119887) (9)




(119910 lowast 119909) are in 119861(119886)













Proof Indeed








(10)

for all 119909 119910 isin 119866



minus1



1199091199100

= 119909 119909 lowast 119910119899+1

= (119909 lowast 119910119899

) lowast 119910 (11)


(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = 0 lowast (119909 lowast 119910)119896 (12)




0 lowast 119909119896+1

= (0 lowast 119909119896

) lowast 119909

= (0 lowast 119886119896

) lowast 119909 ⩽ (0 lowast 119886119896

) lowast 119886 = 0 lowast 119886119896+1

(13)




= 0 lowast (119886 lowast 119887)119896


(0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 0 lowast (119886 lowast 119887)119896 (14)




119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

(15)












3





minus1



sdot 119888minus1

= 119886 sdot 119888minus1

sdot 119887minus1

= (119886 lowast 119888) lowast 119887

(16)











(17)




(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = (0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

= 0 lowast (119886 lowast 119887)119896

= 0 lowast (119909 lowast 119910)119896

(18)





0 lowast (0 lowast 119909119896

) = 0 lowast (0 lowast 119909)119896 (19)







119878 (119860) = ⋃119886isin119860

119861 (119886) = 119909 isin 119866 119909 isin 119861 (119886) 119886 isin 119860 (20)










lowast 0 119886 119887

0 0 0 119887

119886 119886 0 119887

119887 119887 119887 0

(21)

a

b0

(22)








(119909 lowast 119911119898

) lowast (119910 lowast 119911119899

) 119910 isin 119860 997904rArr 119909 isin 119860 (23)








(119909 lowast 119909119898

) lowast (0 lowast 119909119899

) = (0 lowast 119909119898minus1

) lowast (0 lowast 119909119899

) = 0 isin 119860

(24)

















(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ (119909 lowast 119911119899minus2



(25)



119899)lowast(119910lowast119911

119899)) isin 119860





(119909 lowast 119911119899

) lowast (119910 lowast 119911119899






119899


(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)


) lowast (119910 lowast 119911119896

)

(27)





119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)


(28)


119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)



















lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)



3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















= 119861 (119886 lowast 119887) (9)




(119910 lowast 119909) are in 119861(119886)













Proof Indeed








(10)

for all 119909 119910 isin 119866



minus1



1199091199100

= 119909 119909 lowast 119910119899+1

= (119909 lowast 119910119899

) lowast 119910 (11)


(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = 0 lowast (119909 lowast 119910)119896 (12)




0 lowast 119909119896+1

= (0 lowast 119909119896

) lowast 119909

= (0 lowast 119886119896

) lowast 119909 ⩽ (0 lowast 119886119896

) lowast 119886 = 0 lowast 119886119896+1

(13)




= 0 lowast (119886 lowast 119887)119896


(0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 0 lowast (119886 lowast 119887)119896 (14)




119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

(15)












3





minus1



sdot 119888minus1

= 119886 sdot 119888minus1

sdot 119887minus1

= (119886 lowast 119888) lowast 119887

(16)











(17)




(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = (0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

= 0 lowast (119886 lowast 119887)119896

= 0 lowast (119909 lowast 119910)119896

(18)





0 lowast (0 lowast 119909119896

) = 0 lowast (0 lowast 119909)119896 (19)







119878 (119860) = ⋃119886isin119860

119861 (119886) = 119909 isin 119866 119909 isin 119861 (119886) 119886 isin 119860 (20)










lowast 0 119886 119887

0 0 0 119887

119886 119886 0 119887

119887 119887 119887 0

(21)

a

b0

(22)








(119909 lowast 119911119898

) lowast (119910 lowast 119911119899

) 119910 isin 119860 997904rArr 119909 isin 119860 (23)








(119909 lowast 119909119898

) lowast (0 lowast 119909119899

) = (0 lowast 119909119898minus1

) lowast (0 lowast 119909119899

) = 0 isin 119860

(24)

















(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ (119909 lowast 119911119899minus2



(25)



119899)lowast(119910lowast119911

119899)) isin 119860





(119909 lowast 119911119899

) lowast (119910 lowast 119911119899






119899


(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)


) lowast (119910 lowast 119911119896

)

(27)





119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)


(28)


119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)



















lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)



3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












= 0 lowast (119886 lowast 119887)119896


(0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 0 lowast (119886 lowast 119887)119896 (14)




119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

(15)












3





minus1



sdot 119888minus1

= 119886 sdot 119888minus1

sdot 119887minus1

= (119886 lowast 119888) lowast 119887

(16)











(17)




(0 lowast 119909119896

) lowast (0 lowast 119910119896

) = (0 lowast 119886119896

) lowast (0 lowast 119887119896

) = 119886minus119896

sdot 119887119896

= (119886 sdot 119887minus1

)minus119896

= 0 lowast (119886 lowast 119887)119896

= 0 lowast (119909 lowast 119910)119896

(18)





0 lowast (0 lowast 119909119896

) = 0 lowast (0 lowast 119909)119896 (19)







119878 (119860) = ⋃119886isin119860

119861 (119886) = 119909 isin 119866 119909 isin 119861 (119886) 119886 isin 119860 (20)










lowast 0 119886 119887

0 0 0 119887

119886 119886 0 119887

119887 119887 119887 0

(21)

a

b0

(22)








(119909 lowast 119911119898

) lowast (119910 lowast 119911119899

) 119910 isin 119860 997904rArr 119909 isin 119860 (23)








(119909 lowast 119909119898

) lowast (0 lowast 119909119899

) = (0 lowast 119909119898minus1

) lowast (0 lowast 119909119899

) = 0 isin 119860

(24)

















(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ (119909 lowast 119911119899minus2



(25)



119899)lowast(119910lowast119911

119899)) isin 119860





(119909 lowast 119911119899

) lowast (119910 lowast 119911119899






119899


(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)


) lowast (119910 lowast 119911119896

)

(27)





119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)


(28)


119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)



















lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)



3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















lowast 0 119886 119887

0 0 0 119887

119886 119886 0 119887

119887 119887 119887 0

(21)

a

b0

(22)








(119909 lowast 119911119898

) lowast (119910 lowast 119911119899

) 119910 isin 119860 997904rArr 119909 isin 119860 (23)








(119909 lowast 119909119898

) lowast (0 lowast 119909119899

) = (0 lowast 119909119898minus1

) lowast (0 lowast 119909119899

) = 0 isin 119860

(24)

















(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ (119909 lowast 119911119899minus2



(25)



119899)lowast(119910lowast119911

119899)) isin 119860





(119909 lowast 119911119899

) lowast (119910 lowast 119911119899






119899


(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)


) lowast (119910 lowast 119911119896

)

(27)





119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)


(28)


119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)



















lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)



3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)

⩽ (119909 lowast 119911119899minus2



(25)



119899)lowast(119910lowast119911

119899)) isin 119860





(119909 lowast 119911119899

) lowast (119910 lowast 119911119899






119899


(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899minus1


)


) lowast (119910 lowast 119911119896

)

(27)





119910 lowast 119911119899 and

(119909 lowast 119911119899

) lowast (119910 lowast 119911119899

) ⩽ (119909 lowast 119911119899

) lowast (0 lowast 119911119899

)

⩽ (119909 lowast 119911119899minus1


)


(28)


119899) ⩽ 119909 Since (119909 lowast 119911

119899) lowast (119910 lowast 119911

119899) isin 119861(0)



















lowast 0 1 2 3 4 5

0 0 5 0 1 5 1

1 1 0 1 5 0 5

2 2 5 0 1 5 1

3 3 4 5 0 4 2

4 4 2 1 5 0 5

5 5 1 5 0 1 0

(29)

10

4 32

5

(30)



3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3) lowast (0lowast

12) isin 119861(0) and 1 notin 119861(0)




(119909 lowast 119911119899

) lowast (0 lowast 119911119899

) = 0 997904rArr 119909 = 0 (31)






119866 then

0 = (119909 lowast 119911119899

) lowast (0 lowast 119911119899

) ⩽ 119909 lowast 0 = 119909 (32)










= 119886

= 119888 lowast (119887minus1

)minus1

= 119888 lowast 119887 isin 119868 (119860)

(33)











1(119866) = 119861(0)



lowast 0 119886 119887 119888 119889 119890

0 0 0 0 119889 119888 119889

119886 119886 0 119886 119889 119888 119889

119887 119887 119887 0 119889 119888 119889

119888 119888 119888 119888 0 119889 0

119889 119889 119889 119889 119888 0 119888

119890 119890 119888 119890 119886 119889 0

(34)

a b

0

e

c d

(35)








119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896(119866)




for every 119896


119896(119866)


119896= 0 and 0 lowast 119909

119896= 0 lowast 119886

119896= 0 0 lowast 119910

119896=




119861(119886 lowast 119887) sub 119873119896(119866) So119873




119896(119866) Thus 0 lowast 119887119896 = 0 and

0 lowast (119886 lowast 119888)119896

= (0 lowast 119886119896

) lowast (0 lowast 119888119896

)

= ((0 lowast 119886119896

) lowast (0 lowast 119887119896

)) lowast (0 lowast 119888119896

)


= 0

(36)



119896(119866) is a BCC-



algebra


Proof Since119873(119866) = ⋃119896isin119873

119873119896(119866) and 0 isin 119873

119896(119866) for every 119896




= 0 lowast 119887119899= 0



0 = (119886minus119898

)119899

sdot (119887119899

)119898

= 119886minus119898119899

sdot 119887119898119899

= (119886 sdot 119887minus1

)minus119898119899

= 0 lowast (119886 sdot 119887)119898119899

(37)





is nilpotent





Therefore 119866 = 119873(119866)








119896 isin

119860 that is

[119860 119896] = 119909 isin 119866 0 lowast 119909119896

isin 119860 (38)




lowast 0 1 2 3 4 5

0 0 0 0 0 5 5

1 1 0 2 2 4 4

2 2 0 0 0 5 5

3 3 2 2 0 4 4

4 4 5 5 5 0 2

5 5 5 5 5 0 0

(39)

5

2 4

31

0

(40)















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















[119860 119896]




(119909 lowast 119910)119896





=











119896


((0 lowast 119909119896

) lowast ((0 lowast 119911119896

)119898

)) lowast ((0 lowast 119910119896

) lowast ((0 lowast 119911119896

)119899

))

= 0 lowast ((119909 lowast 119911119898

) lowast (119910 lowast 119911119899

)119896

) isin 119860

(41)






(i) antigrouped if

1205932

(119909) isin 119860 997904rArr 119909 isin 119860 (42)

(ii) associative if




(iv) closed if


(v) commutative if











(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(x) obstinate if


(xi) regular if


(xii) strong if


for all 119909 119910 119911 isin 119866





(1205932(119909))119896


1205932

(0 lowast 119909119896

) = 1205932

(0) lowast (1205932

(119909))119896

= 0 lowast (1205932

(119909))119896

isin 119860

(54)



119896





119896


119896



0 lowast (0 lowast 119909)119896

= 0 lowast (0 lowast 119909119896

) isin 119860 (55)


119910)119896


119896



References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on weak-bcc-algebrasalgebras. one of very important identities is the identity ( )...

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