Transcript
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Algebra Universalis, 33 (1995) 68-90 0002 5240/95/010068-23501.50 +0.20/0 �9 1995 Birkhfiuser Verlag, Basel

On the quasivariety of BCK-algebras and its subvarieties

W. J. BLOK AND J. G. RAFTERY

1. Introduction

Most of the familiar algebras of logic, such as Boolean algebras, Heyting algebras and modal algebras, form varieties. In general, however, the equivalent algebraic semantics of a deductive system, algebraizable in the sense of [1], may only be assumed to be a quasivariety. One quasi-identity among those that define the equivalent algebraic semantics takes, in the familiar systems, the form: if x ~ y ~ 1 and y ~ x ~ 1 then x ~ y . In the cases mentioned above, this quasi-iden- tity follows readily from the identities that hold in the equivalent quasivariety. But again, in general, this is not the case, and the quasivariety N E ~ of all BCK-alge- bras is probably the most natural witness of this phenomenon: it is the (unique) equivalent algebraic semantics of a natural and simple deductive system, viz. the "BCK-logic" of Meredith (see [31, p. 316] and [1, 5.2.3]), in which no set of identities may replace the quasi-identity. This paper focusses on the role of the quasi-identity, and in particular on what can be said about the identities that imply it. In model-theoretic terms, therefore, the paper investigates the varieties of BCK-algebras.

We first observe that N E X is the splitting quasivariety associated with a certain 3-element algebra B in a large variety ~ , i.e., ~cgJ t consists of all algebras in V that do not contain a subalgebra isomorphic to B. This perspective will allow us to show that N'~a~ff is the largest subquasivariety of ~ having such desirable proper- ties as being relatively 0-regular and relatively congruence distributive. In view of the fact that N'cgogf itself is not a variety, it also allows us to give a simple proof of Wrofiski's results that the congruences of BCK-algebras do not satisfy any nontriv- ial lattice identities, and are not n-permutable, for any integer n > 2: see [ 40]. This contrasts with the behaviour of the ,,~cg~ff_congruences,, of BCK-algebras A, by which we mean congruences 0 on A such that A/0 s Ncgjl .

Presented by W. Taylor. Received July 30, 1992; accepted in final form June 1, 1993.

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 69

Next we turn our attention to questions relating to what we call BCK-varieties, i.e., varieties contained in Ncgof. Since the congruences of algebras in these varieties coincide with their No'of-congruences, these algebras are congruence distributive and 0-regular. They are not congruence permutable, but Idziak [13] showed that they are congruence 3-permutable. We characterize internally the

algebras A e Ncgof with the property that HS(A) ~_ 9~cgof (which is strictly weaker than HSP(A) ___ 03cr and conclude that the relational square of any tolerance on such an A is a congruence a property strictly stronger than congruence 3-per-

mutability. Komori ' s description of varieties of BCK-algebras and Idziak's result may be seen as corollaries. We prove that the varieties of BCK-algebras form a sublattice of the lattice of subquasivarieties of Ncgof, and that this sublattice, which

is known to have continuum card!nality and no greatest element, contains cofinal chains of order type co. On the other hand, we show that no chain of quasicommu- tative BCK-varieties is cofinal among varieties of BCK-algebras; neither is a natural

chain of varieties arising from Cornish's condition (J). In the process we provide

new examples of BCK-algebras.

2. Algebraic preliminaries

For general universal algebraic background we refer the reader to [3] or [11]. We denote algebras by boldface capitals A, B, C , . . . and their respective universes by A, B, C . . . . . We use co to denote the set of non-negative integers. Let o f be a class of algebras of a given similarity type and A = ( A ; . . . ) a member of of . We

shall make standard use of the class operators I, H, S, P, P, (for subdirect products) and P , (for ultraproducts), while Q(of ) shall denote the quasivariety generated by the class of . A tolerance on A is a reflexive, symmetric binary relation r on A which is also compatible with the fundamental operations of A (i.e., it is also

a subuniverse of A x A). The algebraic lattice of all tolerances on A is denoted by Tol A, while, as usual, Con A is the congruence lattice of A and for a, b e A, OA(a, b) denotes the least congruence 0 on A with (a, b) e 0.

For z, t / ~ A x A, we denote the relational product z o t/ by zt/, and we define zo = ida .'= {(a, a): a E A}, and ~n+ 1 = znz (n ~ co). The least positive n ~ co, if it

exists, such that zn is a congruence for every z e Tol A, is called the tolerance number of A and is denoted by tn(A). We also write t n ( o f ) = n if n is the least positive integer such that tn(B) < n for all B e of . I f tn(A) < n then A is congruence (n + 1)-permutable [34]. The converse fails [4], [32], but a variety o f is congruence (n + 1)-permutable iff tn(o~(() -< n [21], [34].

A congruence 0 of A is called a of-congruence if A/0 e o f ; the set of all of-congruences of A is denoted by C o n ~ A. I f o f is a quasivariety, C o n ~ A is an

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70 W. J. BLOK AND ). G. RAFTERY ALGEBRA UNIV.

algebraic lattice. If the quasivariety oU is fixed and clear from the context, we shall refer to the S-congruences as relative congruences. In this case we say that A is relatively congruence distributive if the lattice C o n x A is distributive, and relatively

O-regular if 0 s A and for all relative congruences 0, 0' of A, 0 = 0' iff 0/0 = 0/0'. We say that A has the relative congruence extension property if for any B e S(A), and any relative congruence 0 of B, there is a relative congruence 0' of A such that 0'c~ (B x B) = 0. We write Lq(•) and PV(Jf) to denote, respectively, the lattice of subquasivarieties and the poset of subvarieties of ~( (ordered by inclusion). Finally,

~sJ shall denote the class of all subdirectly irreducible algebras in a class ~((.

3. Preliminaries concerning BCK-algebras

Is6ki's BCK-algebras (introduced in [14]; see survey articles [7], [18]) arise in the first instance from algebraic logic. In the sense of [ 1], they constitute the equivalent

algebraic semantics for the "BCK-logic" of Meredith (see [31, p. 316]) named after "combinators" B, C and K. This connection with logic is clarified in [1, Section 5.2.3]. Secondly, BCK-algebras are precisely the residuation subreducts of (dually) residuated commutative integral pomonoids [10], [24], [27], [41]. (These structures

are defined and discussed in Section 7.) A BCK-algebra may be defined as an algebra A = <A;- , 0) of type (2, 0) satisfying the following axioms (where the binary operation is abbreviated as justaposition):

((xy)(xz))(zy) ~ o (1)

x0 ~ x (2)

0x ~ 0 (3)

xy ~ 0 ~ yx ~ x ~ y. (4)

The identity (1) is an algebraic version of the B-axiom (y ~ z) ~ ((z ~ x) ~ (y ~ x)) which expresses the transitivity of implication. The quasi-identity (4) corresponds to the one referred to in the Introduction. The relation < on A, defined by a < b iff ab = 0, is a partial order with respect to which 0 is the least element and right (left) "multiplication" by a fixed element of A is isotone (antitone) [18]. Thus the class N~oU of all BCK-algebras satisfies:

xx ~ 0 (5)

xy <- x (i.e., (xy)x ~ 0). (6)

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 71

The identity ( x y ) x ~, 0 is the algebraic version of the axiom K: x --* (y ~ x); it states that an implication is preserved if the premiss is strengthened. Less obviously, ~ c g ~ also satisfies (see [18] for proofs):

x(xy) < y (i.e., (x(xy))y ~ 0) (7)

x(x(xy)) ~ xy (8)

(xy)z ~ (xz)y. (9)

The identity (9) is the algebraic version of the C-axiom which asserts that it is permissible to interchange premisses: (z ~ (y ~ x)) ~ (y ~ (z ~ x)).

The above definition of B C K - a l g e b r a s is more economical than the usual one, which substitutes (5) and (7) for our (2). From (1) and (2) above, one calculates that x x ~ ((x0)(x0))(00)~ 0 and ( x ( x y ) ) y ~ ( (xO)(xy ) ) (yO) . ,~ 0; conversely, (2) is derived from (1), (3), (4), (5) and (7) in [18].

A two-element set {0, a} is the universe of a B C K - a l g e b r a with a0 = a and x y = 0 in all other cases. Conditions (2), (3) and (5) show that there is, up to isomorphism, only one such two-element B C K - a l g e b r a ; we let C denote such an algebra.

In a B C K - a l g e b r a or, more generally, any algebra of type (2, 0), we abbreviate "products" of elements by the inductive rule x y l . . . Yn + 1 '= ( x y l �9 �9 �9 y~)y~ + 1. By an inductive generalization of (9), it follows that ~cgX satisfies x y l . . . Y n ~

xy~(l) �9 �9 �9 y~(n) for any permutation rc of { 1 , . . . , n}; we use the abbreviations:

xII~'= lY i := x y l . . . y~

x y ~ x y " + l , = ( x y " ) y (n > 0 ) .

The following observations are easily provable by induction on the complexity of terms, using (2), (3) and (5).

FACT 0. L e t t = t ( X l , . . . , xn), n > 1, be a t e rm o f t ype (2, 0) /n the var iables

X l , . . . , X n �9

(i) There e x i s t s a g r o u p o i d t e rm s = s ( x l , . . . , x , ) ( f r e e o f occurrences o f O)

such tha t ~r ~ t ~ s. M o r e exp l i c i t l y :

(ii) There e x i s t i ~ { 1 , . . . , n} a n d g r o u p o i d t e r m s u l , �9 �9 �9 Urn, m ~ co, in the

var iab les x~ . . . . . xn ( f r e e o f O) such tha t ~cs ~ t ( x l . . . . . x , ) .~

xirlT= luj(xl . . . . . xm). (iii) I f y ~ { x l , . . . , x , } a n d yi~{0 , y} f o r i = 1 . . . . . n, then ~ c g ~ , ~

t ( y l . . . . . y , ) .,~ 0 or ~cgoU ~ t ( y l . . . . , y~) ~ y . []

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72 W. J. B L O K A N D J. G. R A F T E R Y A L G E B R A U N [ V ,

By its definition, the class ~cgsr is a quasivariety. Wr6nski [39] and Higgs [12] have shown that it is not a variety; its varietal closure is H ( ~ Y ) . The last assertion of Fact 0 amounts to saying that the free BCK-algebra on one free generator is isomorphic to the aforementioned two-element algebra C. By a BCK-variety we mean a variety of algebras of type (2, 0) consisting of BCK-alge- bras. It is welt known that C generates the smallest nontrivial BCK-variety, J . This variety (whose members ar called implicative BCK-algebras [7] or Tarski algebras [23] or implication algebras [22]) consists just of those BCK-algebras that satisfy x(yx) ~ x [7, p. 106]. We may also describe J as the class of all ( . , 0)-subreducts of Boolean algebras (A; A, V , ' , 0 , 1>, where x .y..=x /x (y') [20]. In fact, 0~r is contained in every nontrivial subquasivariety of ~ 'Zs ( , since C embeds into any nontrivial BCK-algebra.

An ideal of a BCK-algebra A is a subset I of A, containing 0, such that a ~ I whenever a e A and b, ab ~ I. The ideals of A form an algebraic distributive lattice, Id A, with respect to set inclusion; the ideal of A generated by X _~ A will be denoted by ( X ) or <X)A. We have <~> = {0}, and for /Z/~ X __ A,

( X ) = { a 6 A : ( 3 n ~ o O ( 3 x l , . . . , x , ~ X ) (ax I . . . x . = O ) } [15]. (lO)

For any z e T o l A , the O-class O / z , = { a e A : ( a , O ) ~ z } is an ideal of A and 0 / rn= O/z for all integers n > 1 [33, Theorem 2.2.b]. On the other hand, an ideal I of A is the 0-class of at least one, and in general, many tolerances on A of which the largest, viz., ~b z ..= {(a, b) ~ A x A : ab, ba ~ I} is a congruence; since A/~b I (writ- ten as A/I) is a BCK-algebra (i.e., satisfies (4)), ~bz is in fact a Meg,-congruence. The maps I ~ q5 t and 0 ~ 0/0 are isotone functions between the lattices Id A, and Con A. Whereas I = 0/~bl for every ideal / , a congruence 0 is generally smaller than

qS0/0 [34], [39], [40].

4. The quasivariety ~

The congruences of BCK-algebras behave quite wildly in general, e.g., they satisfy no special lattice identities and are not n-permutable for any n > 2 [40]. The following proposition shows that the situation is very different where the relative congruences (i.e., the McgJ~ff-congruences) are concerned.

PROPOSITION 1. Let A be a BCK-algebra. (i) The maps 0 ~ (0 ~ C o n ~ A) and I ~ ~z (I ~ Id A) are mutually inverse

lattice isomorphisms between the relative congruence lattice of A and the ideal lattice of A.

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 73

(ii) A is relatively O-regular, relatively congruence distributive and has the relative congruence extension property.

(iii) H ( A ) ~ J f /ff A is O-regular, in which case A is also congruence distributive. I f HS(A) _ ~ g Y , then A has the congruence extension prop- erty.

Proof. (i) In view of earlier remarks, it suffices to show that if 0 ~ C o n ~ , ~ A and I = 0/0 then ~b~ ___ 0. Now (a, b) e 4~1 implies (ab, 0), (ba, 0) E 0, from which it follows, since A[O E ~ c g ~ , that (a, b) e 0 (using (4)).

(ii) Clearly, (i) implies that A is relatively 0-regular. Congruence distributivity follows from Patasifiski's result [25] that every BCK-algebra has a distributive lattice of ideals, while the claim of congruence extensibility follows easily from the fact that if I is an ideal of a subalgebra D of a BCK-algebra E then ( I )E c~ D = I (see [7, Theorem 4.1]).

(iii) follows from (ii), the fact that for any B e N~oU, we have H(B) ___ ~go~f iff Con~e~cB = Con B, and the fact that q~o/0 = 0 for any congruence 0 of a 0-regular BCK-algebra. []

In [30, Question 8], Patasifiski and Wrofiski ask whether the quasivariety NcgJl enjoys the congruence extension property (in the absolute sense). In [45], we showed that this is not the case and that the condition H(A)___ Ncgzf does not imply the condition HS(A) c__ Ncggff.

Let V be the variety defined by the identities (1), (2) and (3), or more generally, any quasivariety of type (2, 0) containing H(NCgou{), satisfying xx ~ O, xO ..~ x and 0x ~ 0, with the additional property that the subquasivariety of ~ defined by (4) is exactly ~)cgy. Let B = (B; -, 0) be the algebra of type (2, 0) with B = {0, a, b}, [B]= 3, and aO=a, bO=b, while in all other cases, xy =0 . Clearly B ~ (4); Wrofiski [39] proves that ~gx( ( is not a variety by showing that B e H(NcgJU).

PROPOSITION 2. The pair (Q(B), N ( ~ ) splits the lattice Lq(3g), i.e., for every quasivariety X ~_ ~ , either oY- ~ N ~ Y " or Q(B) ___ Y (and not both).

Proof. Suppose x ( e Lq("//') and oU q~ N'cgo~ff. From our assumptions on • , it follows that X does not satisfy (4), hence there is an algebra A eoY- and there are elements a, b e A such that ab = 0, ba = 0 but a r b. Then {0, a, b } is the universe of a subalgebra of A isomorphic to B. Hence B ~ X , and Q(B) __ ~ . []

For each n e r o , let B n = ( B , ; ' , 0 ) , where B , = { 0 , al . . . . , a ,} , IBnl= n + l , a i 0 = a i , i = l . . . . . n, and x y = O in all other cases; note that B is iso- morphic to B2. Observe that the subalgebra of (B2)" with universe { ( 0 , . . . , 0 ) } ~ { ( c l . . . . , c , ) : c i v a 0 , i = l . . . . ,n} is isomorphic to B2,. Since

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74 w . J . BLOK AND J. G. RAFTERY ALGEBRA UNIV.

Bj e S(Bk) whenever j < k, it follows that Q(B) contains Bn, n = 1, 2 . . . . . Since H(Bn) = {Bm : m < n} _~ Q(B), every congruence of B n is a Q(B)-congruence. It is easy to see that the congruences of B n other than the universal congruence are in

one-to-one correspondence with the partitions of the set {al . . . . . an }, and that this correspondence preserves the lattice operations as well as the relational product operation. We may conclude:

PROPOSITION 3. (i) Q(B) does not satisfy any nontrivial (relative) congruence identity.

(ii) Q(B) is not congruence n-permutable for any integer n > 2. []

COROLLARY 4. Let ~ be the variety defined by the identities (l), (2) and (3). (i) Ncgof is the largest subquasivariety of "U that is relatively O-regular.

(ii) NcKS is the largest subquasivariety of ~ that is relatively congruence distributive, or that satisfies any nontrivial relative congruence identity.

Proof We have already observed that r162 is relatively 0-regular and rela- tively congruence distributive. Now let X _ ~ be a quasivariety such that X ~cgS((. Then B ~ J~(, and therefore Q(B) ___ ~ff. But Q(B) is not relatively 0-regular (in fact, B itself isn't) and Q(B) does not satisfy any nontrivial relative congruence

identity, by the previous proposition. []

COROLLARY 5. Let 2U ~ N~oU be a quasivariety such that H(~Y') sg ~cgjy'.

Then: (i) s f satisfies no nontrivial congruence identities.

(ii) o~ff is not congruence n-permutable for any n > 2.

Proof. Let ~ be as in the previous corollary. Since H(JY') is a variety with H ( X ) ~ ~ and H ( ~ ) ~ ~(g~f, we may conclude that Q(B) ~ H(X) . Thus, for every algebra D e Q(B), there is an algebra A e X and an epimorphism h : A--* D. By the correspondence theorem, the map 0 ~ h - 1(0) embeds the congruence lattice

of D into that of A, and it is easy to check that this embedding preserves the

relational product as well. []

As a matter of interest, we include the following result.

PROPOSITION 6. The quasivariety Q(B) is not a variety.

Proof. Since B is finite and every nontrivial subalgebra of B is isomorphic either

to B or to the two element BCK-algebra C, we have Q(B) = ISP(B) = IPs{B, C}. Let D be the subalgebra of B x B with universe {(0,0), (a, 0), (0, a),(0, b),

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(a, a), (a, b)}. Let q = idow {(0, a), (0, b)}. It is easily checked that r/~ Con D. Let A = D/r/. Then A ~ HSP(B) but it is straightforward to check that ida is the only congruence of A which does not identify the elements (a, a)/r I and (a, b)/rl. Thus A is not isomorphic to a subdirect product of members of {B, C}. []

5. The conditions H(A) _~ ~ J f and HS(A) _~ ~ J f

In the remainder of the paper, we shall be concerned mainly with BCK-varieties. We have already mentioned that the condition H ( A ) ~ Mcgo~ff is strictly weaker

than H S ( A ) _ ~ c g ~ . Also, H S ( A ) ~ Mc~o~ is strictly weaker than HSP(A)_~ ~c~j~ff. To see this, consider a sequence A1, A2 . . . . of finite BCK-algebras such that no BCK-variety contains all of the Ai (such a sequence is constructed in [42]), and let A be the "direct sum" of these algebras, i.e., the subalgebra of I/ iA i whose elements are just those of II;Ae which are zero in all but finitely many co-ordinates i. Clearly H S P ( A ) ~ ~ c ~ , and A is locally finite. To conclude that HS(A)_~ ~cgX, we need only check that homomorphic images of 2-generated subalgebras of A satisfy the quasi-identity (4), and this is indeed so, since every finite BCK-algebra generates a BCK-variety [5, Theorem 2.2]. We shall give an internal characteriza- tion (Theorem 8) of the condition HS(A) _~ ~cg3~, which generalizes a description of BCK-varieties of Komori and Idziak [1 3].

Let T = (T(x, y); �9 0) be the term algebra (i.e., the absolutely free algebra) of type (2, 0) freely generated by distinct variables x , y and let F = F ~ e ~ ( 2 ) =

(F(ff ,)7), . , 0) be the free BCK-algebra on two (distinct) free generators if, 37. If t e T(x, y), we write /-for tv(~, p) ~ F(~, ~).

Also, let B = (B; �9 0) be as in the previous section and let # : T ~ B be the homomorphism defined by #(x) = a, #(y) = b.

Notice that the twb-element BCK-algebra C may be considered as a subalgebra

of B with universe C = {0, a}. Let c : B-~ C be the homomorphism defined by c(a) = e ( b ) = a . If we also define homomorphisms , ~ : T ~ F and p : F ~ C by

2(x) =if , 2(y)=37 and p( f f )=p(37)= a, we obtain the following commutative diagram:

#

T ~B

F , C P

LEMMA 7. The following conditions are equivalent for a term t = t(x,y) T(x, y):

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76 w . J . BLOK A N D J. G. RAFTERY ALGEBRA UNIV.

(i) F~ ( { ~ , ~7})~; (ii) ~ E X ~ t(x, y ) ( x y ) n ( y x ) m ~ 0 for some n, m e o2;

(iii) # ( 0 = O; (iv) ~<gJ l ~ t(x, x) ~ O.

Proo f (i) => (ii) follows from the characterizat ion (10) of ideals generated by subsets, as well as s tandard properties of free algebras (see [3, Ch. 11, Theorem

1 1.4]). (ii) ~ (iii): Since C ~ M ~ W , it follows f rom (ii) and identities (2) and (5) that

0 = tC(a, a)(aa)~(aa) m = tC(a, a) = e/~(t). Thus #(t) ~ c -1({0}) = {0}.

(iii) =>(iv): Since C ~ ( 6 J l , it follows from (iii) that t C ( a , a ) = c / ~ ( t ) =

c(0) = 0. Therefore ~ ( ~ : ~ t(x, x) ,~ x and so ~(~A5 ~ t(x, x) ~ 0 by Fact 0 (iii). (iv) => (i)" Let J = ( { ~ , y--~))~ and let 0 = O~(~, p). We have (~, 37) ~ ~bs, so

0 _ ~b:. Since F e ~ ( ~ V , it follows f rom (iv) that (t, 0) = (ff(~, 37), t~(~, ~)) ~ 0,

hence (t, 0) ~ ~bj, i.e., Fe J. []

T H E O R E M 8. Let A be any algebra o f type (2, 0) . Then HS(A) ___ ~cgj{" /ff A satisfies (1), (2) and (3) and for each c, d e A, there exist n, m ~ o9 and groupoid

terms ul . . . . . u,, vl . . . . . Vm e T(X, y) with cII7= i u# (c, d) = dl-I~= 1 v# (c, d), such

that ~ c g X ~ u~ (x, x) ,.~ 0 ~ v: (x, x) for i = 1 , . . . , n and j = 1 . . . . . m.

Proo f (=>) Let HS(A)_~ NcgdU. Then A is clearly a BCK-algebra, so A certainly satisfies (1), (2) and (3). Let c, d e A and let r/: T ~ A be the homomor - phism defined by q(x) = c, ~/(y) = d. Since HS(A) ___ #)(gay-, there is no homomor - phism from r/(T) onto the algebra B (defined in Section 4). Since kl is an

epimorphism, it follows from the homomorph i sm theorem that ker(r/) ~ ker(#), so we may choose :t, s E T(x, y) with rl(t) = r/(s) but #(t) r i.e., tA(c, d) = sA(c, d)

but tB(a, b) r sB(a, b). There are now two cases: Case 1: {#(t), #(s)} ~ {{a, 0}, {b, 0}}. Wi thout loss of generality, we will assume

#(t) = a and #(s) = 0. Then N ~ f f ~ t(x, x) ~ x and #)cgJd ~ s(x, x) ~ O, by Lemma

7 and Fact 0 (iii). Let J = ({cd, d c } ) , m . Then (c, d) e q~j, so

C = IA(c, C) ~ j tA(c, d) = sA(e, d) dpe s A ( c , C) = 0 ,

whence c e O/(as = J, and similarly d e J. By (10), there exist k, l, p, q e ~o such that

c(cd)k(dc) ' = 0 = d(cd)P(dc) q.

The result follows if we take n = k + l, m = p + q, u 1 . . . . . U k = U 1 . . . . .

V p = x y and u k + l = ' ' ' = u k + ~ = v p + l = ' = v p + q = y x .

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Case 2: {#(t), kt(s)} = {a, b}. We will assume ~t( t )= a and #(s) = b. By Fact 0

(ii), we m a y assume that we have t = Z U l . . . u , and s = W V l . . . V m for some z, w �9 {x, y}, n, m �9 o) and u~ . . . . . un, v~ . . . . . v,~ �9 T(x, y). By L e m m a 7, # ) E X satisfies t(x, x) ~ x ~ s(x , x), and therefore also ui(x, x) ~ 0 ~ Vj(x, x) (i.e., kt(/~i) = 0 = #(vj)) for i = 1, . . . , n a n d j = 1 . . . . . m. Thus / t (z ) = a and kt(w) = b, so tha t z = x and w = y. N o w the desired result follows f rom tA(c, d) = SA(C, d).

( ~ ) Suppose D is a subalgebra of A and 0 �9 Con D. Then D (and hence D/0) satisfies (1), (2) and (3), and therefore, also (5). To prove that D/0 �9 Ncg~((, we need only show that D/0 satisfies the quasi-identi ty (4). Let c, d � 9 with (cd, 0), (dc, 0) e 0. Choose n, m, u~ . . . . . un, v~ . . . . . Vm as in the s ta tement of the theorem. Write a and b for c/O and d/O, respectively, and suppose a # b. Observe that we m a y identify the subalgebra o f D/0 generated by {a, b} with the three-ele- men t a lgebra B. By L e m m a 7, we have #(u,.) = 0 = #(vj), i.e., u~(a, b) = 0 = v~(a, b) for all i, j , but this implies that a = b, complet ing the proof . []

C O R O L L A R Y 9. I f HS(A)___ Nc .gy then t n ( A ) < 2, hence A is congruence 3-permutable.

Proof. Let z ~ Tol A and (c, d ) ~ z 3. Recall that O/zn= O/z for all posit ive integers n (see [33, Theo rem 2.2b, e]). In part icular , cd, dcEO/'c3=O/z. I f

H S ( A ) _ ~ E X , we m a y choose n, m, u ~ , . . . , un, vl . . . . . Vm as in the s ta tement o f Theo rem 8, and for each i �9 { 1 , . . . , n}, it follows f rom L e m m a 7 that there are k , l � 9 such that uA(c ,d ) ( cd )k (dc ) t=O. Thus ( u A ( c , d ) , O ) � 9 Similarly (v A (c, d), 0) �9 % so

c , erlT= i u# (e, d) = drI5%, v~ (c, d ) , d,

hence (c, d) ~ r 2. This shows that z 3 __~ Z" 2, which means that z 2 �9 Con A, and we

conclude that tn(A) < 2. By [34, Theorem 1.2], A is congruence 3-permutable . []

By contrast , Corol lary 5 shows that an arb i t ra ry BCK-algebra need not be congruence n -pe rmutab le for any integer n > 2 (see [40] also). The following result is stated wi thout p r o o f in [13].

C O R O L L A R Y 10. (Komor i , Idziak) Let ~ be a class o f algebras o f type

(2, 0) . Then HSP(JY') is a BCK-varie ty i f f S satisfies (1), (2), (3) and an identity o f the f o r m

xRT= 1 ui (x, y) ~ ylIT= I vj (x, y) (11)

for some n, m E o9 and some binary groupoid terms Ul . . . . . un, Vl . . . . . v m E T(x, y)

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78 w . J . BLOK AND J. G. RAFTERY ALGEBRA UNIV.

such that NcgX satisfies

u i ( x , x ) ~ O ~ v j ( x , x ) , i = 1 . . . . . n, j = l , . . . , m .

In this case, HSP(X) is congruence 3-permutable.

Proof Sufficiency follows by applying Theorem 8 to any algebra in HSPUf(), since such an algebra will satisfy all identities that hold in ~ff. Necessity and the remaining assertions follow by applying Theorem 8 to the HSP(Jl)-free algebra of type (2, 0) on two free generators (which clearly satisfies HS(A) _c ~cg~). []

Thus BCK-varieties are congruence 3-permutable and since the smallest nontriv- ial BCK-variety J is not congruence permutable, every nontrivial BCK-variety fails to be congruence permutable, whence tn(~) --2. (These facts are quite well known.)

6. The lattice of varieties of BCK-algebras

Recall that Lq(,~c~ff,(") denotes the lattice of all subquasivarieties of ~ c g ~ and that pv(o)cg~) is the partially ordered "set" of all BCK-varieties (both ordered by inclusion). Clearly p~(~cgy) is a meet-subsemilattice of Lq(NcgJl). The authors of [30], [42], in stating order-theoretic properties of p~(Ncg~), appear to have taken for granted that it is a lattice. None of the references that they cite in contexts in which P~(NcgcU) is described as a lattice contain any arguments purporting to establish this fact. In [26, Theorem 1], Patasifiski also asserts that pv(~cg:,,ff) is a lattice, saying that the nontrivial part of the proof follows easily from a result of Yutani [44, Theorem 3] to the effect that the class of all 0-regular BCK-algebras is closed under finite direct products. We do not have a proof along these lines. Here is a different proof.

THEOREM 11. The poset p v ( ~ ( ( ) of all BCK-varieties is a sublattice of the lattice Lq(~icgog(") of subquasivarieties of ~cs Thus P~(~c~X) is an ideal of the lattice L of all varieties of type (2, 0).

Proof. Let ~ , ~ be BCK-varieties. By Corollary 5, there are identities

xlIT- its(x, y) ,~ yIIT= lS~(X, y)

x~In= l Ui(X, y) ~ y l ~ = l Vi(X, y)

ti (x, x) ~ si (x, x) ~ ui (x, x) ~ vi (x, x) ~ 0

(12)

(13)

(14)_

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 79

such that ~1 ~ (12), ~ ~ (13) and ~(goU ~ (14)~, i = 1 , . . . , n. (For notational simplicity we use a uniform n; this loses no generality since ~ c g ~ ~ x ~ x(yy), by (2) and (5).) Now let ~U be the variety of type (2, 0) satisfying the identities (1), (2), (3) and

(x nT_, t~ (x, y))n7_, uj ( xnT= i t, (x, y), y nT=x s~ (x, y))

~,~ (yI 'In= lSi(X, y))I-[7= 1/)j (XI~7= i t~(X, y), yIIT= lsi(x, y)). (15)

Observe that ~)rgjg( satisfies

Uj(xl-In= l ti(x, X), xI'In= lSi(X, x)) ~ uj(x, x) ~ 0

vS(x, x) ~ vj(xIIT= 1 ti(x, x), xIIT= ,si(x, x))

for j = 1 , . . . , n. This, together with the (14); and Corollary 10, implies that ~U is a BCK-variety. Now ~ ~ (15), since (15) is a substitution instance of (13), while in ~1, the left-hand side of (15) reduces, by (12), (2) and the (14);, to

(xFIT= i ti(x, y))rI~= 1 uj (xIIT= ~ ti(x, y), xnT t , ( x , y))

X II n= 1 ti (x, y) II]= 10 ~ x I~n= 1 ti (x, y).

Similarly, the right hand side of (15) becomes yH~.=lSi(x,y) in ~ , so that V 1 ~ (15), as a consequence of (12). Thus ~K1 w~2-~ ~f. Write ~ v ~ for the join of ~1 and V2 in L and ~1 + ~2 for their join in Lq(~Jcg~). Clearly, ~1 v ~ is a BCK-variety.

It follows that ~ v ~2 is a congruence distributive variety (by Proposition l(iii)). As a consequence of J6nsson's Lemma [3, Corollary IV.6.9], (~//] v ~2)si = (~l)SI t.J (~" 2)SI" Thus,

V1 v ~ ~ I S P ( ~ v ~2)si ~ ISP((~/F1)sI w (~2)si) ~ ~ + ~/F2 ~ ~ v ~2 . []

It is known [38] that there are 2 s~ distinct BCK-varieties and that there is no largest BCK-variety [42]. Nevertheless, we note the following result:

COROLLARY 12. The lattice of BCK-varieties contains cofinal chains of order type co.

Proof. The "set" of BCK-varieties with equational bases consisting only of (1), (2), (3) and one identity of the form (11) described in Corollary 10 (where ~ f N r ~ ui(x, x ) ~ 0 ~ vj(x, x) for all i, j ) is cofinal in P~(~JcgX) and may be

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8 0 W. J. B L O K A N D J. G. R A F T E R Y A L G E B R A U N I V .

enumerated, say as ~o, ~//], ~ . . . . . Define ~o = ~o and for each n ~ co, choose ~/C~+~ to be the variety constructed as in the proof of Theorem 11, containing

and ~ + ~ . (Note that ~ + 1 is one of the ~m-) The subsequence ~o, ~/r ~2 . . . . of ~0, ~11, ~ . . . . is a cofinal chain in p~(:~cs

Among the well-studied BCK-varieties, certain countable ascending chains occur naturally. This prompts the question of whether any such easily describable chain is cofinal in P~(N~SC). We shall present two negative results (Propositions 16 and 19) which suggest that this is not the case.

7. Quasicommutative BCK-aigebras and the varieties ~ . , n ~ ~o

Recall that a BCK-algebra is called quasicommutative if it satisfies the identity

(Cim j ): x (xy) i ( yx)J ~ y(xy) m(yx) n

for some i, j, m, n e co. For each choice of i, j, m, n e co, the class of BCK-algebras satisfying (C~J~) is a variety (see [43] or use Corollary 10), which we shall denote by ~ , { ; this class is called a quasi-commutative BCK-variety. The well known variety Y of all commutative BCK-algebras, i.e., BCK-algebras satisfying the identity

(T): x(xy) ~ y(yx),

(introduced by Tanaka [36]), is just Cgo~.l~ For each n E co, the class gn of all BCK-algebras satisfying the identity

(E,): x y n + l ~ x y "

is a quasicommutative BCK-variety, since such algebras also satisfy (C~) [5, Lemma 1.3]. Other identities equivalent to (E,) are mentioned in [7, p. 103] and [5], but the following result, which sharpens [7, Theorem 2.2], seems to have gone unnoticed.

PROPOSITION 13. Let A be a BCK-algebra andre, n ~ co with m > O. Let (R m) be the identity x(x(xy))n~ (x(x(xy))n)y m. Then the following conditions are equiva- lent:

(i) A satisfies (En); (ii) A satisfies (C,"~);

(iii) A satisfies (R'2).

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 81

Thus the identities (1), (2) and (3), together with any one of (E,), ( C ~ ) and (R~) form an equational base for the variety gn, and ~ = cg~ n.

Proof. From (6), the fact that -< is a partial order, and induction it follows that A satisfies (R~) iff A satisfies (R~) for all integers k > 1. Also, any BCK-algebra satisfies xy "+1 < xy ~ (by (6) and the isotonicity of fight "multiplication"); in the presence of (C,","), we also have:

x(x (xy) )" ~ x (x (xy ) )" ( (xy )x )"

(xy)(x(xy))"

(x(x(xy) )")y

(by (6) and (2))

(by (C~n~), (6) and (2))

(by (9)).

i.e., A satisfies (Rn ~) and therefore also (R m) and (R~ + 1). Now A satisfies:

xy" <- x (x (xy ) )"

(x(x(xy) )~ ~ + l

x y n + 1

(by (7) and antitonicity of left multiplication)

(by (R~ + ~))

(by (6) and isotonicity of right multiplication).

Thus A satisfies (En) and the result follows. []

It is known and quite easy to see that every finite BCK-algebra satisfies (En) for some n s co and therefore generates a quasicommutative BCK-variety. The varieties gn have equationally definable principal congruences; in fact, each gn is possessed of a "commutative ternary deductive term" (in the sense of [2]), viz., a term p(x, y, z) such that gn satisfies the identities p ( x , x , z ) ~ z and p(x ,y ,p(x ' , y ' , z)) p(x', y', p(x, y, z)) and for all A ~ E, and a, b, c, d E A, if 0 = O A(a, b) and (c, d) ~ 0 then pA(a, b, c) =pa(a , b, d). In g , , we may take p(x, y, z) = z(xy)"(yx)" [28]. It is also known that if A is a subdirectly irreducible algebra in any of the varieties 8n then the poset (A; < ) has a unique atom [29]. Of course, g0 is the trivial variety; it is known that ~ is locally finite iff n < 1 [8], [9], [19]. The members of ~1 are known as Hilbert algebras [8] or positive implicative BCK-algebras [7], and they are precisely the ( . , 0)-subreducts of dual Brouwerian semilattices (see [2]), i.e., algebras (A; v , . , 0), where (A; v ) is a join semilattice with least element 0 and for a, b e A ,

V x ( a < - b v x .r a .b<_x) .

This relationship between '~1 and Brouwerian semilattices generalizes as follows: if (A; + , O) is a commutative monoid whose identity 0 is the least element of a partial order < on A, compatible with the binary operation + , and for every

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82 w . J . BLOK AND J. G. RAFTERY ALGEBRA UNIV.

a, b E A there is a least element c (denoted a �9 b) of A such that a -< b + c, then we shall call A = (A; + , . , 0) a partially ordered commutative residuated integral monoid or, briefly; a pocrim. Results of Patasifiski [27], Ono and Komori [24] and Fleischer [ 10] prove BCK-algebras to be just the ( - , 0) subreducts of pocrims (see [41] also). This makes pocrims a very useful tool in constructing BCK-algebras, as will be exemplified in Lemma 15. The ( . , 0)-reducts of pocrims have also been called BCK-algebras with condition (S) [16]. Is6ki [17] has shown that they may be axiomatized by (1)-(4) , together with the identity

x(y § z) ~ (xy)z. (16)

Pocrims therefore form a quasivariety and Higgs [12] has shown that, like BCK- algebras, they fail to form a variety, i.e., their homomorphic images need not satisfy (4). Given a pocrim A, we may define, for n E e) and a e A, elements na E A by 0a = 0; (n + 1)a =(na) + a. It is easy to see that an (n + 1)-potent pocrim, i.e., one satisfying

(Mn): (n § 1)x ~ nx,

satisfies (En). Whether every BCK-algebra in ~, is the ( . , 0)-subreduct of an (n + 1)-potent pocrim is an open question raised in [2]. The result is true for n = 1, and simply rephrases the aforementioned relationship between d~ and Brouwerian semilattices (which are precisely the 2-potent pocrims). In general, as an immediate consequence of the next lemma, it would be sufficient to prove that every BCK- algebra in d~ is a ( . , 0)-subreduct of a pocrim satisfying (E~).

LEMMA 14. Let A be a BCK-algebra and let + be a partial binary operation on A such that the equation (16) holds in A whenever its left-handside is defined. I fn ~ ~o and A ~ gn then the equation (Mn) holds in A whenever its left-hand side is defined.

Proof Assume that (n + 1 ) x = ( n x ) + x exists in A. Using (16), (3), (5) and (En), we calculate:

(nx)((n + 1)x) ~ ((nx)(nx))x ~ 0;

((n + 1)x)(nx) ~ ((n + 1)x)x" ~ ((n + 1)x)x "+l

((n + 1)x)((n + l)x) ~ 0,

and applying (4), we deduce (M,). []

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 83

All of the BCK-varieties mentioned thus far are quasicommutative, and there

are clearly many countable ascending chains of quasicommutative varieties, e.g., the sequence d~ n e co, is a strictly ascending chain [7, p. 107]. In [6], Cornish studied

the class j of all BCK-algebras satisfying

(J) : x ( x ( y ( y x ) ) ) ~ y (y(x(xy) ) ) ,

which is a BCK-varie ty (again this follows from Corollary 10) and which properly

contains the join of the varieties d~ and J-. We shall now show that J is not contained in any quasicommutative BCK-variety; thus, no chain of quasicommuta- five BCK-varieties is cofinal in PV(sScgf).

LEMMA 15. Let 0 < k ~ co and for each a ~ co, let qa, ra ~ co be the unique

integers such that a = k q , + r, and O < r~< k. Let 0 , "--, and �9 be the binary

operations on co defined by:

a O b ,=k(qa + qb) + min{ra + rb, k - 1}

a -" b .'= max{a - b, 0}

a �9 b := (a + (rb "-- ra)) "-- b

= ~k(qa - qb) + (r~ "-- rb) i f qb < qa ;

~ 0 i f qa < qb.

(i) A~- ..= (co; G, �9 0) is a pocrim, the partial order being the natural order <

on co; hence

(ii) A k , = ( c o ; ' , 0) is a BCK-algebra, whose associated partial order is the

natural order < on co.

(iii) A k satisfies (J).

(iv) A l is the simple commutative linearly ordered BCK-algebra (co; "-, 0) .

(v) I f k > 1 then Ik := {a ~ co : a < k } is the unique nontrivial ideal o f Ak ; Ik is a

subalgebra o f A~, and Ag/Ik ~ AI .

(vi) I f k > 1 then Ak satisfies none o f the quasicommutative identities ( C ~ ) .

Proo f (ii) follows immediately from (i). The proofs of (i) and (iii) are tedious but routine case checking exercises and will be omitted, while (iv) is trivial and (v) is easily verified. To prove (vi), let k > 1 and i, j m , n ~co. If we set x = k j + k - 1

and y k ( j + 1) then the left and right hand sides of i j = (Cmn) take the values, respectively, of k - 1 and k ( ( j + 1) -" n), which cannot be equal. []

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84 W. J. BLOK AND J. G. RAFTERY ALGEBRA UNIV.

This construction seems to be new, though A 2 w a s used in [23] as an example of a non-commutative BCK-algebra which is "tolerance trivial", i.e., it has toler- ance number 1, and the subalgebra {0, 1, 2, 3) of A 2 w a s considered by Wozni- akowska [37].

Clearly A k is commutative only for k = 1, but (v) implies that each Ak is a

member of the "Mal'cev square" g- o J - of the variety of commutative BCK-alge-

bras. (g- o J - is the class of all algebras A = (A; �9 0) of type (2, 0) such that for

some 0 ~ C o n A , the class 0/0 is the universe of a subalgebra B of A with A/0, B ~ Y. ) Although (iii) and the fact that J - _ j might suggest a close relation- ship between J" o ~-- and J , these classes are incomparable, in view of an example of Seto [35], [6, pp 341-342] of a simple noncommutative BCK-algebra in the variety J , and our forthcoming example in Lemma 18: this is a BCK-algebra

D r J , having an ideal S which is the universe of a commutative subalgebra of D such that D / S is isomorphic to the two-element BCK-algebra C. From (iii) and (vi) we obtain the following result immediately.

PROPOSITION 16. The variety J is not contained in the union o f all quasicom-

mutative BCK-varieties. Thus, no chain o f quasicommutative BCK-variet ies is cofinal in the lattice p v ( ~ ) . []

8. The varieties ~ . , n ~ co

There is a sequence of natural "generalizations" of Cornish's condition (J). For each n 6 { - 1 } • co, we define, inductively, a term Jn (X, y) by

J _ l ( X , y ) = x

j2n(x, Y) = Y(Y(J2n- l(X, y))),

J2, +1 (x, y) = x (x ( j2 , (x, y))),

and an identity:

(J,): L(x , y) ~L(Y, x).

Thus (J_ l) defines the trivial variety, (J0) is the identity distinguishing the commu- tative BCK-algebras within ~ c g ~ and (J1) is just (J). In the next result we use the standard abbreviation t(x, y, ~) for a term t whose variables are among x, y, zl , . . . , z k for s o m e k ~ c o .

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 85

LEMMA 17. (i) Every BCK-algebra satisfies jn+ l(x, y) < jn(x, y) and L + l ( x , y ) < _ L ( y , x ) for all n e { - 1 } w c 0 . Hence, i f a BCK-algebra satisfies L + l(X, y) ~ j , ( x , y), it must also satisfy (J~). In particular, every finite BCK-algebra is a member of J , for some n > - 1 .

(ii) I f a BCK-algebra A satisfies an identity o f the form xl-I~'= l ui(x,y, ~) y I I ; = 1G(x, Y, ~) (e.g., if A is a member of some BCK-variety) then A also satisfies xHim lu~(x, y, ~) <- ~(x, y) for all n e { - 1} u~o.

(iii) The class fin of all BCK-algebras satisfying (J,) is a variety for each n e { - 1 ) um.

(iv) The sequence J - l , J O , J l , J 2 , . . . , is a strictly ascending chain in P ~ ( ~ ) .

Proof (i) By (7), j ~ + l ( x , y ) < j ~ ( x , y ) for all n ~ { - 1 } u o , and j o ( x , y ) = y(yx) < y = j l(y, x) by (6). Now for n z ~ ,

J2n + 1 (x, y) = x(x( j2 . (x, y))) -< x(x ( j2 ._ , (y, x))) = J2. (y, x);

J2. + 2(x, y) = y(y(J2. +1 (x, y))) < y(y(J2 . (y , x))) =Jzn + 1 (y , x),

using the isotonicity (antitonicity) of right (left) multiplication, and induction hypotheses. The second assertion follows from the first using the symmetry of the variables. The last assertion follows from the others, simply by noting that the poset underlying a finite BCK-algebra satisfies the descending chain condition.

(ii) The claim is trivial for n = - 1 . For n ~ co,

(yHkvDjzn(X, y) ~ ( y (A , (x , y)))Hkvk

= (Y(Y(Y( • , -1 (x, y))))) 1-Ikv k

(Y(J2, 1( x, Y)))Hkvk

( y IIkvk )j2, - 1 ( X, y)

~ 0

(by (9))

(by (8))

(by (9))

(by an induction hypothesis)

and similarly (xIIiui)jzn+ l (x, y ) ~ O, as required. (iii) An easy inductive argument shows that for each n ~ {-1}wee, there are

terms t,(x, y) and sn(x, y) such that {L(x, Y) ,L(Y , x)} = {x(t,(x, y)), y(s,(x, y))} and N c g X ~ tn(x, x) ~ 0 ~ s,(x, x). By Corollary 10, each J , is a BCK-variety.

(iv) Certainly J - 1 --- J o c J l (both inclusions proper: see [6]), while in Jan 1, where n > 0,

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86 W. J. BLOK AND J. G. RAFTERY ALGEBRA UNIV.

j2,,(x, y) = Y( Y(A,, - 1.( x, Y))) ~ Y( Y(.L,,-1 (Y, x)))

= Y(Y(Y(Y(J2n -- 2(Y, X))))) ,~ y(y(.~,, _ 2(Y, X)))

=J~, - I(Y, x) ~ J 2 , - l(x, y) ~J2,,(Y, x)

(by (8))

(by symmetry)

so J2n-1 - - - J2n , and similarly, J2n c J2n+l - To show that these inclusions are strict, we refer to the sequence Dn, 0 < n ~ co, of finite BCK-algebras constructed by Wrofiski and Kabzifiski in [42]. Each D, is generated by its two maximal elements

ao and b0. It is easily checked that for each n > 0, Dn satisfies (an ~) but not ( J , -2 ) :

indeed { L - 2(ao, bo), L-2(b0 , ao) } = {a~_ l, b ,_ i }. []

Corollary 12 and Lemma 17 (iv) tempt one to speculate that the chain of

varieties i n , n E co, is cofinal in P~(~fgoff). We show, however, that this is not the case. (Elementarily describable cofinal chains of order type co in P~(~cgS) seem to

be quite elusive.) We need the following construction. Let S be the set of all rational numbers of the form n/2 m, where n, m ~ ~o. Let

A = {a s :s ~ S} and B = {bs :s ~ S} be S-indexed sets such that the functions

s ~ as and s ~ bs are one-to-one, ao = b0 r S and the sets A\{ao}, B\{bo} and S are mutually disjoint. Let D = A w B w S, and define a binary operation �9 (abbreviated

by juxtaposition) on D by the following rules, where i , j ~ S:

aiaj = bibj -- aibj = ij = i =j== max{0, i - j }

b~aj = i - ( j / 2 )

aij = ai= j

bij = b i - j

iaj = ibj = 0.

LEMMA 18. (i) D = ( D ; ' , 0 ) is a BCK-algebra. (ii) The associated partial order < on D (defined by x < y iff xy = O) has the

following properties, where i ,L k ~ S (a portion o f the Hasse diagram of (D; -< ) is

depicted in Figure 1):

i <aj; i <bk;

aj < bk iff j <_ k;

bj < ak iff j < k /2.

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Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 87

b 4

o.# k 3

~ 62

d f~

f

x I

~0 -- ~0

; t

I

6 O

Figure 1

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88 w.J. BLOK AND S. ~. RAva-zr~v AL~EB~ t3N~V.

(iii) D satisfies the quasicommutative identity

(C332): x (xy )3 (yx ) 2 ,,~ y (xy )2 (yx ) 3.

(Thus, D is a member o f the BCK-varie ty cg2332.)

(iv) For each n ~ { - 1 } ~ c o , D fails to satisfy the identity (J,).

Proo f (i) Axiom (1) is a tedious case checking exercise; the rest is easy, as is

(ii).

(iii) I t is easy to see that the subalgebras o f D whose universes are A u S and

B ~ S each satisfy the identity x ( x y ) 2 ( y x ) ~ y ( x y ) ( y x ) 2, which implies (C~3Z). I f

(x, y) ~ (A x B ) w ( B x A), one checks that both sides o f (C 32) take the value ao.

(iv) One proves by induction on n e co that

j 2 n _ , D ( a , , b l ) ~ A \ { a o } , j z , ~

j R . _ l ~ ~ B\{ao}, j2 .O(bl ,a l ) ~ A\(ao}. []

The last result is an immediate consequence o f this lemma.

P R O P O S I T I O N 19. The BCK-varie ty (g32~ is not contained in the union o f the

varieties J . , n ~ co. Thus the chain o f varieties J . , n ~ co, is not cofinal in the lattice

P~ []

Acknowledgements

The second author thanks the Depar tment o f Mathematics, Statistics and

Compute r Science at the University o f Illinois at Chicago for its hospitality during

the Fall Semester o f 1991. Both authors thank the participants in the Universal

Algebra Seminar at U I C for helpful discussions relating to this work, and Teo

Sturm for a contr ibut ion to the construct ion in Lemma 15.

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Department of Mathematics, Statistics and Computer Science

University of Illinois at Chicago 851 South Morgan Street Chicago, Illinois 60607-7045 USA

Department of Mathematics and Applied Mathematics

University of Natal, Pietermaritzburg Private Bag X01 Scottsville 3209 South Africa


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