birkhoff centre of a poset

Southeast Asian Bulletin of Mathematics (2002) 26: 509–516 Southeast AsianBulletin ofMathematics: Springer-Verlag 2002

Birkho¤ Centre of a Poset

U.M. Swamy, G.C. Rao, R.V.G. Ravi Kumar, and Ch. Pragati

Department of Mathematics, Andhra University Visakhapatnam—530 003, India

AMS Subject classification (1991): 06A12, 20M15

Abstract. In this paper, it is proved that the Boolean centre of a semigroup S with su‰-ciently many commuting idempotents is isomorphic to the inverse limit of the directedfamily of Birkho¤ centres (or Boolean centres) of a class of bounded semigroups. TheBirkho¤ centre is defined for any poset and proved that it is a relatively complementeddistributive lattice whenever it is nonempty. It is observed that for a semilattice S, theBirkho¤ centres as a semigroup and as a poset coincide. Also it is observed that for a Lat-tice ðL;5;4Þ, the Birkho¤ centres of the semilattices ðL;5Þ and ðL;4Þ coincide with theBirkho¤ centre of L. Finally it is proved that for a lattice ðL;5;4Þ, the Boolean centres ofthe semilattices ðL;5Þ and ðL;4Þ coincide with the Boolean centre of L.

Keywords: Birkho¤ centres of a poset and semigroup, Direct factor, Balanced direct factor,Boolean centre, Inverse limit

Introduction

In [6], U.M. Swamy and G.S. Murti introduced the concept of the Boolean centreBðAÞ of a universal algebra A as the set of all balanced direct factors of A whichadmit a balanced direct complement and also proved that it is a permutable Boo-lean sublattice of the lattice CðAÞ of all congruences on A. In [5], Swamy andMurti introduced the concept of the Birkho¤ centre BðSÞ of a semigroup S with 0and 1 and proved that it is a Boolean algebra in which the meet operation is theoperation in S. This concept is extended to a more general semigroup S with suf-ficiently many commuting idempotents (s.m.c) (see definition 1.2) in [3] by G.S.Murti and proved that it is a relatively complemented distributive lattice in whichthe meet operation is the operation in S (provided BðSÞ is nonempty). It is observedin [6] that the Boolean centre BðSÞ of a semigroup S with 0 and 1 is isomorphicto the Birkho¤ centre BðSÞ of S. Whereas in the case of a semigroup S with s.m.c,the Boolean centre BðSÞ need not coincide with the Birkho¤ centre BðSÞ of S. Inthis paper we prove that the Boolean centre BðSÞ of a semigroup S with s.m.c isisomorphic to the inverse limit of the directed family of the Birkho¤ centres (orBoolean centres) of a class of intervals of S each of which is a subsemigroup of Swith 0 and 1.

It is well known that the centre of a bounded partially ordered set (poset) is aBoolean algebra in which the operations are l.u.b and g.l.b in P [1]. We call this asthe Birkho¤ centre BðPÞ of the poset P. In this paper, we extend the above conceptfor a general poset P and prove that BðPÞ is a relatively complemented distributivelattice in which the operations are l.u.b and g.l.b in P (provided BðPÞ is non-empty). In the case of a semilattice S, we observe that its Birkho¤ centres as asemigroup and as a poset coincide. Also we observe that for a lattice ðL;5;4Þ, theBirkho¤ centres of the semigroups ðL;5Þ and ðL;4Þ coincide with the Birkho¤centre of L. Finally we prove that for a lattice ðL;5;4Þ, the Boolean centres of thesemilattices ðL;5Þ and ðL;4Þ coincide with the Boolean centre of the lattice L.

1. Birkho¤ Centre of a Semigroup

In this section, we prove that the Boolean centre of a semigroup S with s.m.c isisomorphic to the inverse limit of the directed family of Birkho¤ centres (or Boo-lean centres) of a class of intervals of S each of which is a subsemigroup (of S)with 0 and 1. We begin with the following.

A semigroup S with 0 and 1 is a semigroup having elements 0 and 1 such that0x ¼ 0 ¼ x0 and x1 ¼ x ¼ 1x for all x A S. An element e of a semigroup S iscalled a commuting idempotent if e2 ¼ e and ex ¼ xe for all x A S. A commutativesemigroup in which every element is idempotent is called a semilattice. In a semi-lattice ðS; :Þ, if we define a relation a by aa b, ab ¼ a, then a is a partial orderon S with respect to which for any a; b A S, a:b ¼ Inf sfa; bg. It is easy to verifythat the set EðSÞ of all commuting idempotents of a semigroup S is a subsemigroupof S, which is also a semilattice. Also for any e; f A EðSÞ such that ea f , theinterval ½e; f �s :¼ fx A S j ex ¼ e and fx ¼ xg is a subsemigroup (of S) with e as0 and f as 1.

Let us recall that a congruence y on a universal algebra A is called a direct fac-tor of A if there is a congruence F on A such that yVF ¼ DA and y �F ¼ A� A;F is called a direct complement of y. A congruence y on a universal algebra A iscalled balanced if ðy4FÞV ðy4F 0Þ ¼ y for all direct factors F and its directcomplements F 0. The set BðAÞ of all balanced direct factors of A which admitbalanced direct complements is called the Boolean centre of A. This BðAÞ is apermutable Boolean sublattice of the lattice CðAÞ of all congruences on A [6].

Definition 1.1 [5]. An element a of a semigroup S with 0 and 1 is called a central

element of S if there exist semigroups S1 and S2 with 0 and 1 and an isomorphism of

S onto S1 � S2 that maps the element a onto ð1; 0Þ. The set BðSÞ of all central ele-ments of S is called the Birkho¤ centre of S.

In [5], it is proved that the Birkho¤ centre of a semigroup S with 0 and 1is a Boolean algebra in which the meet operation is the operation in S andthat a congruence y on S is a direct factor of S if and only if y ¼ ya :¼fðx; yÞ A S � S j ax ¼ ayg for some a A BðSÞ. Further it is observed that a 7! y 0a isan isomorphism of BðSÞ onto the Boolean algebra of all direct factors of S. Hence

510 U.M. Swamy, G.C. Rao, R.V.G. Ravi Kumar, and Ch. Pragati

every direct factor of S is balanced and therefore the Boolean centre BðSÞ is pre-cisely the set fya j a A BðSÞg of all direct factors of S.

Definition 1.2. Let S be a semigroup in which (i) EðSÞ is directed above, that is,

whenever e; f A EðSÞ there exists g A EðSÞ such that e; f a g and (ii) to each x A S

there exist e; f A EðSÞ such that ea f and x A ½e; f �s. Then S is said to be a semi-

group with su‰ciently many commuting idempotents (semigroup with s.m.c).

Here afterwards S denotes a semigroup with s.m.c. It is easy to verify that ifx1; x2; . . . xn A S and n is a positive integer then there exist e; f A EðSÞ such thatea f and xi A ½e; f �s for 1a ia n.

Definition 1.3 [3]. An element a of S is called a central element if there exist semi-

groups S1 with 1 and S2 with 0 and an isomorphism of S onto S1 � S2 that maps

a onto ð1; 0Þ. The set BðSÞ of all central elements of S is called the Birkho¤ centre

of S.

In [3], it is proved that an element a of S is central element if and only if abelongs to the Birkho¤ centre B½e; f �s of the bounded semigroup ½e; f �s whenevere; f A EðSÞ such that ea f and a A ½e; f �s and that BðSÞ is a relatively com-plemented distributive lattice in which the meet operation is the operation in S

(provided BðSÞ is nonempty). Further it is proved that BðSÞ is isomorphic to asublattice of the Boolean centre BðSÞ of S. As in the case of a semigroup with 0and 1, it is not necessary that BðSÞ is isomorphic to BðSÞ; for, if L is a relativelycomplemented distributive lattice then L is a semigroup with s.m.c and BðLÞ ¼ L,whereas the Boolean centre BðLÞ is always a Boolean algebra. Now we prove thatthe Boolean centre BðSÞ of S is the set of all direct factors of S. First we prove thefollowing.

Lemma 1.4. Let y be a direct factor of S with a direct complement y 0 and e; f A EðSÞwith ea f and T ¼ ½e; f �s. Then yV ðT � TÞ is a direct factor of T and its direct

complement is y 0 V ðT � TÞ.

Proof. Let a ¼ yV ðT � TÞ and a 0 ¼ y 0 V ðT � TÞ. Then aV a 0 ¼ DT is clear. Letx; y A T . Then there exists z A S such that ðx; zÞ A y 0 and ðz; yÞ A y. We haveðex; ezÞ A y 0 and ðez; eyÞ A y and hence ðe; ezÞ A yV y 0 ¼ Ds and therefore e ¼ ez.Also we have ð fx; fzÞ A y 0 and ð fz; fyÞ A y and hence ðx; fzÞ A y 0 and ð fz; yÞ A y

and fz A T . Therefore ðx; yÞ A a � a 0 and hence a is a direct factor of T and a 0 is itsdirect complement. 9

Lemma 1.5. Every direct factor of S is balanced.

Proof. Let y and a be direct factors of S and a 0 a direct complement of a.Let ðx; yÞ A ðy4aÞV ðy4a 0Þ so that there exist elements x ¼ x0; x1; . . . xn ¼ y andx ¼ y0; y1; . . . ym ¼ y in S such that ðxi; xiþ1Þ A yU a and ðyj ; yjþ1Þ A yU a 0

for 0a ia n� 1 and 0a jam� 1. Choose e; f A EðSÞ such that ea f and

Birkho¤ Centre of a Poset 511

xi; xj A ½e; f �s for 0a ia n and 0a jam. Let T ¼ ½e; f �s, F ¼ yV ðT � TÞ,b ¼ aV ðT � TÞ and b 0 ¼ a 0 V ðT � TÞ. Then by the above lemma, F, b, b 0 aredirect factors of T and b 0 is a direct complement of b. Since T is a semigroupwith 0 and 1, F, b, b 0 are balanced. Since ðxi; xiþ1Þ A yU a and ðyj ; yjþ1Þ A yU a 0

for 0a ia n� 1 and 0a jam� 1 and xi; yj A T for 0a ia n and 0a jam,it follows that ðx; yÞ A ðF4bÞV ðF4b 0Þ ¼ FJ y. Hence ðy4aÞV ðy4a 0ÞJ yand therefore y is balanced. 9

As a consequence of the above lemmas, we have the following.

Theorem 1.6. The Boolean centre of S is precisely the set of all direct factors of S.

Our next step is to prove that the Boolean centre BðSÞ is the inverse limit of thedirected family of Boolean algebras. First we recall the following.

A pre-ordered set ðI ;aÞ is a non empty set I equipped with a reflexive andtransitive binary relation a. A directed set is a pre-ordered set ðI ;aÞ which isdirected above, that is, for i; j A I , there exists k A I such that i; ja k. A directedfamily of algebras of type t is a triplet ðAi;Fi; j; IÞ where (i) I is a directed pre-ordered set. (ii) For each i A I , Ai is an algebra of type t. (iii) Fi; j is a homo-morphism of Aj into Ai for all ia j such that Fi; j �Fj;k ¼ Fi;k whenever ia ja k

and Fi; i is the identity morphism on Ai for all i A I .An inverse limit of a directed family ðAi;Fi; j; IÞ of algebras of type t is a pairðA; fFigi A I Þ satisfying (i) A is an algebra of type t. (ii) For each i A I , Fi is ahomomorphism of A into Ai such that Fi; j �Fj ¼ Fi whenever ia j. (iii) IfðA 0; fF 0igi A I Þ is a pair satisfying (i) and (ii) above, then there exists a unique mor-phism F of A 0 into A such that Fi �F ¼ F 0i for all i A I .

It is well known that in the category of universal algebras of a given type, theinverse limit of a directed family of algebras exists and is unique upto isomor-phism; infact, for any directed family ðAi;Fi; j; IÞ of universal algebras of type t,

A :¼ a AQi A I

Ai j for ia j;Fi; jðajÞ ¼ ai

� �where ai is the i-th projection of a for all

i A I , and this is denoted by lim ðAi;Fi; j ; IÞ.

Let Q ¼ f½e; f �s j e; f A EðSÞ; ea f g. Then Q is a poset under the relation setinclusion J. Also Q is directed above. For each I A Q, I is a subsemigroup of S

with 0 and 1 and hence the Birkho¤ centre BðIÞ and Boolean centre BðIÞ areisomorphic. For I J J in Q, define FI ;J : BðJÞ ! BðIÞ by FI ;JðyÞ ¼ yV ðI � IÞ.Then by lemmas 1.4 and 1.5, it follows that FI ;J is well defined and pre-serves complements and intersections. Therefore FI ;J is a Boolean algebra homo-morphism. Also it is easy to verify that FI ; I is the identity map on BðIÞ andFI ;J �FJ;K ¼ FI ;K whenever I ; J;K A Q and I J JJK and hence

�BðIÞ;FI ;J ; Q

�is a directed family of Boolean algebras. Let A be the inverse limit of

�BðIÞ;FI ;J ; Q

�.

Then A :¼ y AQi A I

BðIÞ j for I J J in Q; yJ V ðI � IÞ ¼ yI

� �where yI is the pro-

jection of y in BðIÞ. Observe that for I ; J A Q, I J J, y A A implies yI J yJ . Nowwe prove the following.


Lemma 1.7. Let y A A and F ¼ 6I A Q

yI . Then F A BðSÞ and for each I A Q,FV ðI � IÞ ¼ yI .

Proof. For x A S, there exists I A Q such that x A I and hence ðx; xÞ A yI JF.Let ðx; yÞ A F. Then ðx; yÞ A yI for some I A Q and hence ðy; xÞ A yI JF. Letðx; yÞ; ðy; zÞ A F. Then ðx; yÞ A yI and ðy; zÞ A yJ for some I ; J A Q. Choose K A Qsuch that I ; JJK . Then ðx; yÞ; ðy; zÞ A yK and hence ðx; zÞ A yK JF. Letðx; yÞ A F and z A S. Then ðx; yÞ A yI and z A J for some I ; J A Q. Choose K A Qsuch that I ; JJK . Then ðx; yÞ A yK and z A K and hence ðxz; yzÞ; ðzx; zyÞ AyK JF. Therefore F is a congruence on S. Let ðyI Þ0 be a direct complement of yIso that y 0I ¼ ðyI Þ

0 and y 0 A A. Let F 0 ¼ 6I A Q

ðyI Þ0. Then F 0 is a congruence on S. Let

ðx; yÞ A FVF 0. Then ðx; yÞ A yI and ðx; yÞ A ðyJÞ0 for some I ; J A Q. Choose K A Qsuch that I ; JJK . Then ðx; yÞ A yK V ðyKÞ0 ¼ DK and hence x ¼ y and thereforeFVF 0 ¼ Ds. Let x; y A S. Then there exists I A Q such that x; y A I . Choose z A I

such that ðx; zÞ A yI and ðz; yÞ A ðyI Þ0. Then ðx; zÞ A F and ðz; yÞ A F 0 and henceðx; yÞ A F 0 �F and therefore F 0 �F ¼ S � S. Thus F is a direct factor of S witha direct complement F 0 and hence F A BðSÞ. Let I A Q and ðx; yÞ A FV ðI � IÞ.Then there exists J A Q such that ðx; yÞ A yJ . Choose K A Q such that I ; JJK .Then ðx; yÞ A yK V ðI � IÞ ¼ yI and hence FV ðI � IÞ ¼ yI . 9

Lemma 1.8. For any y;F A BðSÞ, yV ðI � IÞJFV ðI � IÞ for all I A Q if and only

if yJF.

Proof. Suppose that y;F A BðSÞ and yV ðI � IÞJFV ðI � IÞ for all I A Qand ðx; yÞ A y. Choose I A Q such that x; y A I . Then ðx; yÞ A yV ðI � IÞJFVðI � IÞJF and hence yJF. Converse is clear. 9

Now, we prove that the Boolean centre of S is isomorphic with the inverse limitA of the directed family

�BðIÞ;FI ;J ; Q

�.

Theorem 1.9. BðSÞ is isomorphic with lim

�BðIÞ;FI ;J ; Q

�.

Proof. If y A BðSÞ then by lemma 1.4, yV ðI � IÞ A BðIÞ for all I A Q. Now definea : BðSÞ ! A by aðyÞI ¼ yV ðI � IÞ for all y A BðSÞ and I A Q. For I ; J A Q andI J J we have aðyÞJ V ðI � IÞ ¼ yV ðJ � JÞV ðI � IÞ ¼ yV ðI � IÞ ¼ aðyÞI andhence a is well defined. By the above lemmas 1.7 and 1.8, it follows that a is anorder isomorphism. 9

2. Birkho¤ Centre of Poset

In this section, we define the Birkho¤ centre BðPÞ for any poset P and prove that itis a relatively complemented distributive lattice in which the operations are l.u.b.and g.l.b in P. Also we observe that for a semilattice S which is directed above, theBirkho¤ centres of S as a semigroup with s.m.c and as a poset coincide. Further


we observe that for a lattice ðL;5;4Þ, the Birkho¤ centres of the semigroups ðL;5Þand ðL;4Þ coincide with the Birkho¤ centre of the lattice L considered as a posetunder the induced order. We recall the following.

Definition 2.1 [1]. An element a of a bounded poset P is called a central element of P

if there exist bounded posets P1 and P2 and an order isomorphism of P onto P1 � P2

such that a is mapped onto ð1; 0Þ. The set BðPÞ of all central elements of P is called

the Birkho¤ centre of P.

It is well known that the Birkho¤ centre BðPÞ of a bounded poset P is a Booleanalgebra in which the operations are l.u.b. and g.l.b. in P. In the case of a boundedlattice L, the Birkho¤ centre BðLÞ is the set of all neutral and complemented ele-ments of L [1]. Now we define the Birkho¤ centre for any poset.

Definition 2.2. An element a of a poset P is called a central element of P if there

exist posets P1 with 1 and P2 with 0 and an order isomorphism P onto P1 � P2 such

that a is mapped onto ð1; 0Þ. The set BðPÞ of all central elements of P is called the

Birkho¤ centre of P.

Note that the Birkho¤ centre BðPÞ of a poset P may be empty. For, a poset Pin which the elements are incomparable, BðPÞ is empty. In case of BðPÞ is nonempty, we prove that BðPÞ is a relatively complemented distributive lattice inwhich joins and meets are respectively l.u.b and g.l.b in P. We have the following.

Lemma 2.3. Let ðP;aÞ be a poset and a A P. Then a A BðPÞ if and only if a5x

and a4x exist for all x A P and the mapping x 7! ða5x; a4xÞ is an order

isomorphism of P onto ða�P � ½aÞP. Where ða�P ¼ fx A P j xa ag and ½aÞP ¼fx A P j aa xg.

Proof. Suppose that a A BðPÞ. Then there exist posets P1 with 1 and P2 with 0and an order isomorphism a : P! P1 � P2 such that aðaÞ ¼ ð1; 0Þ. Let x A P andaðxÞ ¼ ðx1; x2Þ where x1 A P1, x2 A P2. Choose e; f A P such that aðeÞ ¼ ðx1; 0Þand að f Þ ¼ ð1; x2Þ. Then it is easy to verify that e ¼ a5x and f ¼ a4x andhence the mapping F : x 7! ða5x; a4xÞ of P into ða�P � ½aÞP is well defined andorder preserving. Let x; y A P and aðxÞ ¼ ðx1; x2Þ, aðyÞ ¼ ðy1; y2Þ and a5xa

a5y and a4xa a4y. Then a ða5xÞa a ða5yÞ and a ða4xÞa a ða4yÞand hence x1 a y1 and x2 a y2 and therefore xa y. Now we prove that F is onto.Let s; t A P such that sa aa t and aðsÞ ¼ ðs1; s2Þ aðtÞ ¼ ðt1; t2Þ. Then s2 ¼ 0 andt1 ¼ 1. Choose x A P such that aðxÞ ¼ ðs1; t2Þ. Then a5x ¼ s and a4x ¼ t andhence F is onto and therefore F is an order isomorphism. Converse is clear. 9

Lemma 2.4. If a; b; e A BðPÞ and aa ea b, then e has a complement e 0 in ½a; b�Pand e 0 is in BðPÞ.

Proof. Suppose that a; b; e A BðPÞ and aa ea b. Then by the above lemma, thereexists e 0 A P such that e 05e ¼ a and e 04e ¼ b and hence e 0 is a complement of


e in ½a; b�P. Choose posets P1 with 1 and P2 with 0 and an order isomorphisma : P! P1 � P2 such that aðeÞ ¼ ð1; 0Þ. Let aðe 0Þ ¼ ðe1; e2Þ. Then aðaÞ ¼ ðe1; 0Þand aðbÞ ¼ ð1; e2Þ. By the above lemma and the fact that a; b A BðPÞ, it followsthat e 05x and e 04x exist for all x A P and the mapping x 7! ðe 05x; e 04xÞ isan order isomorphism of P onto ðe 0�P � ½e 0ÞP and hence e 0 A BðPÞ. 9

The following theorem follows from the above two lemmas.

Theorem 2.5. Let P be any poset and BðPÞ nonempty. Then BðPÞ is a relatively

complemented distributive lattice in which the operations are l.u.b and g.l.b in P.

In the case of a directed poset P (that is, a poset in which any two elementshave an upper bound and a lower bound), it is known from [3], that the Birkh-o¤ centre BðPÞ coincides with the set CðPÞ :¼ fa A P j a A ½x; y�P; x; y A P impliesa A B½x; y�Pg where B½x; y�P is the Birkho¤ centre of the bounded poset ½x; y�P ¼fa A P j xa aa yg. If we omit the directedness of P, then BðPÞ may not coincidewith CðPÞ. Infact, CðPÞ need not be a lattice at all. For, the poset P ¼ fa; b; 1gwith a < 1, b < 1, CðPÞ ¼ P which is not a lattice. Whereas BðPÞ ¼ f1g. If L is alattice, then L is a directed poset under the induced ordering and it is easy to verifythat the Birkho¤ centre BðLÞ is the set of all neutral and relatively complementedelements of L. We state a lemma whose proof is direct verification.

Lemma 2.6. Let A;B be nonempty sets each with a binary operation. Then A;Bare semilattices if and only if A� B is a semilattice under the componentwise

operation.

Let S be a semilattice which is directed above with respect to the induced order.Then S is a semigroup with s.m.c and by the above lemma, it follows that theBirkho¤ centres of S as a semigroup and as a poset coincide. If ðL;5;4Þ is a lat-tice, then ðL;5Þ and ðL;4Þ are semilattices which are directed above and the par-tial orders induced by 5 and 4 are dual to each other and hence the Birkho¤centres of ðL;5Þ and ðL;4Þ coincide with the Birkho¤ centre BðLÞ of L. We provethe same result for Boolean centres in the next section.

3. Boolean Centre of a Lattice

In [6], it is proved that the Boolean centre of a lattice ðL;5;4Þ is precisely the setof all direct factors of L and that a congruence y on a bounded lattice L is directfactor of L if and only if y ¼ ya :¼ fðx; yÞ A L� L j a5x ¼ a5yg for some neu-tral and complemented element a of L. In this section we prove that for a latticeðL;5;4Þ, the Boolean centres of the semilattices ðL;5Þ and ðL;4Þ coincide withthe Boolean centre of the lattice L. We have the following.

Lemma 3.1. Let ðL;5;4Þ be a bounded lattice and y be a direct factor of the semi-

lattice ðL;5Þ. Then y is a direct factor of the lattice L.


Proof. It is enough if we prove that y is a lattice congruence also. Since y isa direct factor of the semilattice ðL;5Þ, there exist a A L such that y ¼ ya :¼fðx; yÞ A L� L j a5x ¼ a5yg. Also this element a is neutral and complemented.In particular, a5ðx4zÞ ¼ ða5xÞ4ða5zÞ for any x; z A L. Now ðx; yÞ A y

and z A L imply that a5x ¼ a5y and hence a5ðx4zÞ ¼ ða5xÞ4ða5zÞ ¼ða5yÞ4ða5zÞ ¼ a5ðy4zÞ and therefore ðx4z; y4zÞ A ya ¼ y. Thus y is alattice congruence. 9

We remark that, in the above lemma, if y is a direct factor of the semilatticeðL;4Þ then y is a direct factor of the lattice L. Now we prove the above lammawhen boundedness of L is omitted.

Lemma 3.2. Let ðL;5;4Þ be a lattice and y be a direct factor of the semilatticeðL;5Þ. Then y is a direct factor the lattice L.

Proof. It is enough if we prove that y is a lattice congruence also. Let x; y; z A L

and ðx; yÞ A y. Choose a; b A L such that aa b and x; y; z A I :¼ ½a; b�L. Thenby lemma 1.4, yV ðI � IÞ is a direct factor of the bounded sub semilattice I ofðL;5Þ. Hence by the above lemma, yV ðI � IÞ is a direct factor of I and ðx; yÞ AyV ðI � IÞ and z A I . Which imply that ðx4z; y4zÞ A yV ðI � IÞJ y. Hence y isa lattice congruence. 9

We remark that, in the above lemma, if y is a direct factor of the semilatticeðL;4Þ then y is a direct factor of the lattice L. The following theorem is a conse-quence of the above lemma and the fact that the Boolean centre of the semilatticeðL;5Þ or ðL;4Þ is precisely that set of all direct factors of ðL;5Þ or ðL;4Þ respec-tively.

Theorem 3.3. Let ðL;5;4Þ be a lattice. Then the Boolean centres of the semilattices

ðL;5Þ and ðL;4Þ coincide with the Boolean centre of the lattice L.

References

1. Birkho¤, G.: Lattice Theory, American Mathematical Society, Colloquium Publications,25, 1967.

2. Jacobson, N.: Basic Algebra, Vol. II, Hindustan Publishing Corporation (India), 1994.3. Murti, G.S.: Boolean Centre of a Universal algebra, Doctoral thesis, Andhra University,

Waltair, 1980.4. Stanely, B., Sankappanavar, H.P.: A course in Universal algebra, Springer-Verlag, 1981.5. Swamy, U.M., Murti, G.S.: Boolean Centre of a semigroup, Pure and Applied Mathe-

matika Sciences 13(1–2), 1981.6. Swamy, U.M., Murti, G.S.: Boolean Centre of a Universal Algebra, Algebra Universalis

13, 202–205 (1981).


birkhoff centre of a poset

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