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<ul><li><p>Algebra Universalis, 33 (1995) 68-90 0002 5240/95/010068-23501.50 +0.20/0 9 1995 Birkhfiuser Verlag, Basel </p><p>On the quasivariety of BCK-algebras and its subvarieties </p><p>W. J. BLOK AND J. G. RAFTERY </p><p>1. Introduction </p><p>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 NE~ 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. </p><p>We first observe that NEX is the splitting quasivariety associated with a certain 3-element algebra B in a large variety ~, i.e., ~cgJt 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 &gt; 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. </p><p>Presented by W. Taylor. Received July 30, 1992; accepted in final form June 1, 1993. </p><p>68 </p></li><li><p>Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 69 </p><p>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. </p><p>2. Algebraic preliminaries </p><p>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. </p><p>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 tn (o f )= n if n is the least positive integer such that tn(B) &lt; n for all B e of. I f tn(A) &lt; n then A is congruence (n + 1)-permutable [34]. The converse fails [4], [32], but a variety of is congruence (n + 1)-permutable iff tn(o~(() -&lt; n [21], [34]. </p><p>A congruence 0 of A is called a of-congruence if A/0 e of ; the set of all of-congruences of A is denoted by Con~ A. If o f is a quasivariety, Con~ A is an </p></li><li><p>70 W. J. BLOK AND ). G. RAFTERY ALGEBRA UNIV. </p><p>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 Conx 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 ~((. </p><p>3. Preliminaries concerning BCK-algebras </p><p>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 = </p></li><li><p>Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 71 </p><p>The identity (xy)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, ~cg~ also satisfies (see [18] for proofs): </p><p>x(xy) &lt; y (i.e., (x(xy))y ~ 0) (7) </p><p>x(x(xy)) ~ xy (8) </p><p>(xy)z ~ (xz)y. (9) </p><p>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)). </p><p>The above definition of BCK-a lgebras is more economical than the usual one, which substitutes (5) and (7) for our (2). From (1) and (2) above, one calculates that xx ~ ((x0)(x0))(00)~ 0 and (x (xy) )y ~ ((xO)(xy))(yO). ,~ 0; conversely, (2) is derived from (1), (3), (4), (5) and (7) in [18]. </p><p>A two-element set {0, a} is the universe of a BCK-a lgebra with a0 = a and xy = 0 in all other cases. Conditions (2), (3) and (5) show that there is, up to isomorphism, only one such two-element BCK-a lgebra ; we let C denote such an algebra. </p><p>In a BCK-a lgebra or, more generally, any algebra of type (2, 0), we abbreviate "products" of elements by the inductive rule xy l . . . Yn + 1 '= (xy l 9 9 9 y~)y~ + 1. By an inductive generalization of (9), it follows that ~cgX satisfies xy l . . .Yn ~ xy~(l) 9 9 9 y~(n) for any permutation rc of {1 , . . . , n}; we use the abbreviations: </p><p>xII~'= lY i := xy l . . . y~ </p><p>xy~ xy"+l ,=(xy" )y (n &gt;0) . </p><p>The following observations are easily provable by induction on the complexity of terms, using (2), (3) and (5). </p><p>FACT 0. Let t = t (X l , . . . , xn), n &gt; 1, be a term o f type (2, 0) /n the variables Xl , . . . ,X n 9 </p><p>(i) There ex is ts a groupo id term s = s (x l , . . . , x , ) ( f ree o f occurrences o f O) such that ~r ~ t ~ s. More expl ic i t ly: </p><p>(ii) There ex is t i ~ {1, . . . , n} and groupo id terms ul , 9 9 9 Urn, m ~ co, in the variables x~ . . . . . xn ( f ree o f O) such that ~cs ~t (x l . . . . . x , ) .~ </p><p>xirlT= luj(xl . . . . . xm). (iii) I f y~{x l , . . . , x ,} and yi~{0, y} fo r i=1 . . . . . n, then ~cg~,~ </p><p>t (y l . . . . . y , ) .,~ 0 or ~cgoU ~ t (y l . . . . , y~) ~ y. [] </p></li><li><p>72 W. J. BLOK AND J. G. RAFTERY ALGEBRA UN[V , </p><p>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&gt;, 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. </p><p>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 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]. </p><p>4. The quasivariety ~ </p><p>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 &gt; 2 [40]. The following proposition shows that the situation is very different where the relative congruences (i.e., the McgJ~ff-congruences) are concerned. </p><p>PROPOSITION 1. Let A be a BCK-algebra. (i) The maps 0 ~ (0 ~ Con~ A) and I ~ ~z (I ~ Id A) are mutually inverse </p><p>lattice isomorphisms between the relative congruence lattice of A and the ideal lattice of A. </p></li><li><p>Vol. 33, 1995 On the quasivariety of BCK-algebras and its subvarieties 73 </p><p>(ii) A is relatively O-regular, relatively congruence distributive and has the relative congruence extension property. </p><p>(iii) H(A)~J f /ff A is O-regular, in which case A is also congruence distributive. I f HS(A) _ ~gY, then A has the congruence extension prop- erty. </p><p>Proof. (i) In view of earlier remarks, it suffices to show that if 0 ~ Con~,~ 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 ~cg~, that (a, b) e 0 (using (4)). </p><p>(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]). </p><p>(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. [] </p><p>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. </p><p>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). </p><p>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). </p><p>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) __ ~. [] </p><p>For each nero , let Bn=(B, ; ' ,0 ) , where B ,={0, al . . . . ,a,}, IBnl= n+l , a i0=ai , i= l . . . . . n, and xy=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 iv a0 , i= l . . . . ,n} is isomorphic to B2,. Since </p></li><li><p>74 w. J . BLOK AND J. G. RAFTERY ALGEBRA UNIV. </p><p>Bj e S(Bk) whenever j &lt; k, it follows that Q(B) contains Bn, n = 1, 2 . . . . . Since H(Bn) = {Bm : m &lt; 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: </p><p>PROPOSITION 3. (i) Q(B) does not satisfy any nontrivial (relative) congruence identity. </p><p>(ii) Q(B) is not congruence n-permutable for any integer n &gt; 2. [] </p><p>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 </p><p>distributive, or that satisfies any nontrivial relative congruence identity. </p><p>Proof We have already observed that r162 is relatively 0-regular and rela- tively congruence distributive. Now let...</p></li></ul>