Varieties of Commutative Residuated Integral Pomonoids and Their Residuation Subreducts

JOURNAL OF ALGEBRA 190, 280᎐328Ž. 1997 ARTICLE NO. JA966834

Willem J. Blok*

Department of Mathematics, Statistics and Computer Science, Uni¨ersity of Illinois, Chicago, Illinois 60607 View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector

James G. Raftery†

Department of Mathematics and Applied Mathematics, Uni¨ersity of Natal, King George V A¨enue, Durban 4001 Communicated by Leonard Lipshitz

Received April 27, 1995

INTRODUCTION

The notion of residuation can already be found in Dedekind’s work on modules, and it has played an important role in ideal theory ever since. If R is a commutative ring with 1, and I and J are ideals of R then the residual of I with respect to J is the ideal commonly denoted I: J and defined as Ä4x g R: xJ : I . That is, I: J is the ideal characterized by the condition KJ : I iff K : I: J. This operation is well defined on the collection of all ideals of R and serves to capture the concept of division in the ring in terms of its ideals. Indeed, if R is an integral domain and i,jgRare such that i s kj for some element k g R, then Ž.Ž.i : j s Žk .; for r g R, Ž.r stands here for the ideal generated by r. Krullwx Kru24 and Ward and Dilworthwx WD39 started a long line of investigation showing that much of the structure theory of Noetherian rings can be obtained in the abstract setting of lattices with a suitable multiplication operation Ž.Žabstracted from ideal multiplication and residuation abstracted from ideal residuation as described above. .

*E-mail address: [email protected]. † E-mail address: [email protected].

280

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. VARIETIES OF RESIDUATED POMONOIDS 281

Residuation also plays a central role in the algebraic study of logical systems. Typically the algebras that arise from logic are partially ordered by a relation reflecting deducibility, and endowed with operations realizing the connectives and quantifiers of the logical system. The fundamental connection between conjunction n Ž.usually a semilattice operation and ª is given by

c n b F a iff c F b ª a Ž.) for all elements a, b, c in the universe of the algebra. In the algebraic models of classical propositional logic, the Boolean algebras, the condition leads to the usual definition of b ª a s Ž.! b k a, while in the models of intuitionistic propositional logic the condition is used to define the implication in terms of the partial order and the conjunction. In the models of many valued logics and linear logic the conjunction is just a commutative operation that respects the partial order; here again the implication is defined in terms of the conjunction via Ž.) . In the structures referred to above the crucial elements are the partial order and two binary operations, one of which is a commutative monoid operation [ that respects the partial order, the other one the residuation y˙ ; these three are related by

a F c [ b iff a y˙ b F c. ŽFor technical reasons we use a formulation dual to the one used in the two classes of example described above.. The aim of the present paper is to begin an algebraic investigation of classes of such structures ² A; [ , y˙ ; F :. In doing this we will assume in addition that the structure is integral, i.e., the unit 0 of the monoid operation is the least element in the partial order. This assumption is satisfied in most examples from the literature. For instance, in the monoid of ideals of a commutative ring with 1, the largest ideal R plays the role of the unit, whereas in the algebras of logic an element T representing ‘‘the true’’ does the same. The assumption allows us to eliminate the partial order from the type of the structures Ž.since now a F b iff a˙᎐b s 0 and to treat them as proper algebras, which will be termed pocrims. In an attempt to characterize a suitable notion of residuation without reference to a monoid operation and partial order, Iseki´´ proposedwx Ise66 a set of axioms inspired by a little known logical system that had been introduced by Meredith. The models of the set of axioms are called BCK-algebras after the traditional labels of the combinators B, C, and K attached to the axioms of the logical system, and they have been the subject of intense investigation. It is easily verified that the residuation subreducts of pocrims are BCK-algebras. That Iseki’s´ axioms do indeed 282 BLOK AND RAFTERY characterize the notion of residuation was proved in the mid-1980s independently by Pałasinskiwx Pał82 . Ono and Komori w OK85 x , and Fleischer wxFle88 , who showed that, conversely, every BCK-algebra is a subreduct of a pocrim. Knowledge of this fact has facilitated the solution of some previously open problems concerning BCK-algebras, e.g., it allows one to reduce them to problems concerning pocrims, which generally are better behaved and easier to handle. The class M of all pocrims is a quasivariety which, by a result of Higgs, is not a variety. Our focus in this paper is mainly on varieties of pocrims. ŽThese varieties form the algebraic counterparts of strongly algebraizable logics with conjunction and implication, in the sense ofwx BP89 .. We establish some properties enjoyed by all such varieties, and we study in some detail special classes of varieties, in particular the varieties of n-potent pocrimsŽ. for some n g ␻ and, at the other side of the spectrum, the varieties of cancellative pocrims. We pay special attention to the classes of BCK-algebras that arise as the subreducts of the varieties of pocrims considered. We start by noting, in Sections 2 and 3, global properties of the quasivariety of all pocrims: it is relatively congruence distributive, has the relative congruence extension property, and is relatively point regular with respect to the monoid identity 0. Subvarieties of M, being 0-regular, are congruence n-permutable for some n. We give an equational scheme characterizing pocrim varieties, in which the number of equations coincides with the degree of permutability. Nevertheless, it seems a hard problem to determine whether all such varieties are in fact 3-permutable. Whereas no nontrivial variety of BCK-algebras is congruence permutable, manyŽ. but not all pocrim varieties are. A sufficient condition for permutability is given, which is applicable to all permutable pocrim varieties known to us. A related and apparently hard problem asks whether the quasivariety of residuation subreducts of a pocrim variety must always itself be a variety. We characterize pocrim varieties with this property syntactically, showing in particular that they are 3-permutable. In Section 4, we give examples of pocrim varieties, indicating among them several well-known varieties of classical algebra or logic. We show that the pocrim varieties such that for some n g ␻, all members have n-potent monoid operations are precisely the varieties of pocrims with equationally definable principal congruencesŽ. EDPC . This extends earlier results about varieties of hoopsŽ. in the sense of Buchi¨ and Owens and Wajsberg algebras. Similarly, a variety of BCK-algebras has EDPC iff for some n, it satisfies x y˙˙ny f x yŽ.n q 1 y. The variety of BCK-algebras defined by this identity is known as Enn; we denote by M the variety of Ž. nq1 -potent pocrims. The residuation subreduct class of Mn is a subvariety of En. A plausible conjecture influencing some of our subsequent VARIETIES OF RESIDUATED POMONOIDS 283

investigations is that it exhausts En; we obtain partial confirmation of this in Section 6. We prove that all subquasivarieties of En are varieties; the same is not true of Mn. In Section 5, we examine the construction of ordinal sums of pocrims. Many subvarieties of M are closed under this construction. In several familiar varieties, such as the variety of Brouwerian semilattices and the variety of hoops, the finitely generated subdirectly irreducible algebras in the variety can be obtained from a finite number of simple algebras in the variety by a finite number of applications of the operations of variety generation and ordinal sum, and this fact has profound implications for the structure theory of these varieties. We show that within any variety of pocrims closed under ordinal sums, any variety V gives rise to a variety V q that is also closed under ordinal sums, the structure of which is closely tied to that of V . In particular, if V is locally finite or generated as a quasivariety by its finite members then so is V q. These facts generalize results fromwx BF concerning hoops. We give our construction new application in Section 6, by examining its s s effect on the varieties Mnnand E generated, respectively, by all simple Ž. nq1 -potent pocrims and all simple members of En. Here, in addition to congruence distributivity and closure under ordinal sums, we have avail- s able the theory of varieties with EDPC, from which it follows that Mn and s s Ennare semisimple; in fact, we show that M is a discriminator variety. sq sq Thus, the subdirect irreducibles of Mnnor E are ordinal sums of which the first summand is a simple algebra in Mnnor E . We use this fact to sq sq sq obtain equational axiomizations of Mnnand E , and to prove that M n is congruence permutable. By contrast, we show that Mn is not permutable sq unless n s 1. As a further application, we prove that En is exactly the sq residuation subreduct class of Mn , evidence in favour of the conjectured corresponding relationship between Ennand M . In Section 7, we give special consideration to the case n s 2; we prove sq sq that M22is locally finite. The same is true of E , in view of the aforementioned relationship, and contrasts with Dyrda’s result that neither sq M22nor E is locally finite. We show that M3is not locally finite. Since Mn is a variety of finite type with EDPC, any finite subdirectly irreducible Ž. nq1 -potent pocrim A is splitting in Mnn, i.e., M has a largest subvariety not containing A. Extending this property, we prove that M22and E each /0 have 2 semisimple subvarieties, whereas M11and E Žthe varieties of Brouwerian semilattices and Hilbert algebras. each have just one semisimple subvarietyŽ the varieties of generalized Boolean algebras and Tarski algebras. . Pocrims whose underlying monoids are cancellative are examined in Section 8 and lie at the opposite end of our topic’s spectrum to n-potent pocrims, since for each n, only the trivial cancellative pocrim is n-potent. 284 BLOK AND RAFTERY

Cancellative pocrims form a subquasivariety C of M which is not a variety and which lacks the congruence extension propertyŽ. in the absolute sense . Nevertheless, C has a Mal’cev term and therefore generates a congruence permutable varietyŽ. not contained in M , enabling us to prove that the subvarieties of C form a lattice L¨ Ž.ŽC the corresponding question for M being open. . Our focus in Section 8 is primarily on properties of this lattice, whose structure seems at present quite elusive. We show that L¨ Ž.Chas a unique atom: the smallest nontrivial variety of cancellative pocrims is generatedŽ. even as a quasivariety by the pocrim of nonnegative integers, with the natural addition and order. We identify a denumerable strictly ascending chain in L¨ Ž.C , but we do not know whether L¨Ž.C is countable nor, indeed, whether it is linearly ordered. The main result of this section is that L¨ Ž.C has no greatest element.

1. ALGEBRAIC PRELIMINARIES

For general universal algebraic background we refer the reader to wxwxwBS81 , Gra79¨ , or MMT87 x . We denote algebras by boldface capitals A, B, C, . . . and their respective universes by A,B, C,... . We use ␻ to denote the set of nonnegative integers. Let K be a class of algebras of a given similarity type and A s ²:A; . . . a member of K. We shall make standard use of the class operators I, H, S, P, Ps Ž.for subdirect products , and Pu Ž.for ultraproducts . A tolerance on A is a reflexive, symmetric binary relation ␶ on A which is also compatible with the fundamental operations of A Ž.Ž.i.e., it is also a subuniverse of A = A . The algebraic lattice of all tolerances on A is denoted by Tol A, while, as usual, Con A is the congru- A A ence lattice of A. For X : A, TXŽ.and ⌰ Ž.X denote, respectively, the least tolerance and the least congruence on A that contain X; we abbrevi- A A A ate TaŽÄŽ.Ž.11,b,..., ann,b 4.by TaŽŽ11,b .,..., Žann,b .. and TaŽŽ,b .. by TaAŽ.,b, and we apply similar conventions to ⌰ A. The following characterization of finitely generated tolerances will be needed.

LEMMA 1.1wx Cha81 . Let A be an algebra and a1,...,an,b1,...,bn,c, Ž. AŽŽ . Ž .. d g A. Then c, d g Ta11,b,..., ann,b iff there is a 2n-ary polynomial G on A such that

c s GaŽ.Ž.1,...,an,b1,...,bn and d s Gb1,...,bn,a1,...,an.

For ␶ , ␩ : A = A, we usually denote the relational product ␶ (␩ by ␶␩, 0 ÄŽ. 4 nq1 nŽ. and we define ␶ s idA [ a, a : a g A and ␶ s ␶␶ ng␻. The n least positive n g ␻, if it exists, such that ␶ is a congruence for every ␶gTol A, is called the tolerance number of A and is denoted by tnŽ.A .We also write tnŽ.K s n if n is the least positive integer such that tn Ž.B F n VARIETIES OF RESIDUATED POMONOIDS 285 for all B g K.If tnŽ.A F n then A is congruence Žn q 1 . -permutable wxRS92 , i.e., for any two congruences ␪, ␸ of A, the Ž.n q 1 -fold relational products ␪␸␪␸ иии and ␸␪␸␪ иии coincide. The converse failsŽ e.g.,wx Cha88. , but a variety K is congruence Ž.n q 1 -permutable iff tn Ž.K F n wRS92, Kea93x . We drop the prefix from ‘‘n-permutable’’ when n s 2. A congruence ␪ of A is called a K-congruence if Ar␪ g K; the set of all K-congruences of A is denoted by Con K A.If Kis a quasivariety, Con K A is an algebraic lattice. If the quasivariety K is fixed and clear from the context, we shall refer to the K-congruences as relatië congruences. In this case we say that A is relatiëly congruence distributië if the lattice Con K A is distributive, and relatiëly 0-regular if 0 g A and for all relative congruences ␪, ␪ Ј of A, ␪ s ␪ Ј iff 0r␪ s 0r␪ Ј. We say that A has the relatië congruence extension property if for any B g SAŽ.and any relative congruence ␪ of B, there is a relative congruence ␪ Ј of A such that ␪ Ј l Ž B = B.Žs␪. We drop ‘‘relative Ž. ly ’’ if K is known to be a variety. . We use LqŽ.ŽKresp. P¨Ž..K to denote the latticeŽ resp. the poset. of subquasivari- etiesŽ. resp. subvarieties of a quasivariety K, order by inclusion, and we replace P¨ Ž.K by L¨Ž.K if P¨Ž.K is known to be a lattice, e.g., if K is a variety. The class of all subdirectly irreducible algebras in a variety K will be denoted by KSI . We insist that subdirectly irreducibleŽ in particular, simple. algebras be nontrivial. A congruence permutable, congruence distributive variety is called arithmetical.

2. POCRIMS AND THEIR RESIDUATION SUBREDUCTS

Let ²:A; [ ,0;F be a commutati¨e Ž. dually integral pomonoid, i.e, ²:A;Fis a partially ordered set, ²A; [, 0 : is a commutative monoid whose identity 0 is the least element of ²:A; F , and F is compatible with the monoid operation [ in the sense that a [ b F c [ d whenever a, b, c, d g A with a F c and b F d. If for each a, b g A there is a least element c Ž.denoted a y˙ b of A such that a F c [ b, we say that ²:Ž.A;[,0;F is dually residuated and we call y˙ the residuation operation. The resulting structure A s ²:A; [, y˙ ,0;F will then be called a partially ordered commutati¨eŽ. dually residuated Ž. dually integral monoid or, briefly, a pocrim. In this case it follows that for any a, b, d g A we have a y˙ b F d iff a F d [ b; thus d [ b is the greatest element of Äe g A: e y˙˙b F d4, while a F b iff a y b s 0. Consequently, the partial order F is determined by the monoid operation [ and the residuation operation y˙ , so that no harm comes of systematically confusing a pocrim A with the AAA underlying algebra ² A; [ , y˙ , 0:Ž . We drop the superscripts whenever there is no danger of confusion.. In this spirit, we may say that the class of all pocrims, which we denote by M, is a quasivariety of algebras of type 286 BLOK AND RAFTERY

²:2, 2, 0 , since Iseki´´wx Ise80 has shown that pocrims may be axiomatized by the following set of four identities and a quasi-identity:

Ž.Ž.Ž.Ž.x y˙˙˙y y x y z y ˙˙z y y f 0,Ž. 1 xy˙0fx,2Ž. 0y˙xf0,Ž. 3 Ž.xy˙˙y yzfxy ˙ Ž.z[y,4 Ž. Ž.Ž.xy˙˙yf0& yyxf0ªxfy.5Ž. It is easy to deduce that every pocrim satisfies the following identities:

x y˙x f 0,Ž. 6 Ž.Ž.x[y y˙y [yfx[y.7Ž.

It also follows easily that in the poset ²:A; F underlying a pocrim A, any doubleton Ä4a, b has the element a y˙˙Ž.a y b as a lower bound and the element Ž.a y˙ b [ b as an upper bound. The set of ideals of a commutative ring R with 1 is a pocrim with respect to the monoid operation of ideal multiplication and theŽ. lattice order of re¨ersed set inclusion. Residuation is defined by I y˙ J s I: J s Ä x g R: xJ : I4. A further simple example of a pocrim is the set ␻ with natural addition and the natural linear order, where residuation is defined by ay˙ bsmaxÄ4 0, a y b . Another is the set of all subsets of a given set, with union as the monoid operation, set inclusion as the partial order, and set difference as residuation. If A is a pocrim and a g A, we may define, inductively, elements na g A Ž.ng␻by the following rules: 0a s 0; Ž.Ž.n q 1 a s na [ a. It also follows fromŽ. 4 that pocrims satisfy

Ž.x y˙˙y y z f Ž.x y ˙˙z y y.8Ž. FromŽ. 1 and Ž. 8 , we obtain

Ž.Ž.Ž.Ž.x y˙˙˙y y z y y y ˙˙x y z f 0.Ž. 9

If we abbreviate Ž.x y˙˙y y z as x y ˙˙y y z then expressions such as a y˙ b1y˙˙иии ybnmay be interpreted unambiguously, and for each n g ␻,we use x y˙˙ny as an abbreviation of a ²:y, 0 -term defined inductively by x y ˙ 0ysx;xy˙˙˙Ž.nq1ysxyny y y. It is easy to deduce from the definitions that every pocrim satisfies

Ž.Ž.Ž.Ž.x [ y y˙˙˙z [ w F x y z [ y y w .10 Ž. VARIETIES OF RESIDUATED POMONOIDS 287

For any element c of a pocrim, the unary polynomial function determined by x y˙˙c Ž.Žresp. c y x is isotone resp. antitone . . It turns out that the class of all ²:y˙ , 0 -subreducts of pocrims is precisely the quasivariety BCK of all BCK-algebras, which was introduced by Iseki´´ inwx Ise66 : see survey articleswx IT78 and w Cor82a x . This relationship is discussed in detail by Wronski´ inwx Wro85 and depends on an embedding theorem published by Pałasinski´ wx Pał82 , Ono and Komori wxOK85 , and Fleischer wx Fle88 . The ²:y˙ , 0 -reducts of pocrims have also been called BCK-algebras with condition Ž.Swx Ise79´ . Henceforth if L is either a class of algebras ²:²:A; [ , y˙ , 0 of type 2, 2, 0 or a class of algebras ²:A; y˙˙, 0 of type ²: 2, 0 , we denote the ²:y, 0 -reduct of A g L by Ayyand we write L for ÄBy: B g L 4. We therefore have that BCK s SŽ M y. and that a universal first-order sentence in the language ²:y˙ , 0 holds in BCK iff it holds in M. We shall need to make use of the construction used by Fleischer, Ono, and Komori in the proof of the result BCK s SŽ M y., so we present a version of it here. Let A s ²:A; y˙ , 0 be any BCK-algebra with associated poset ²:A; F . ªªŽ. For each finite sequence a s a1,...,an of elements of A, let Ja s Ž.Ä4Ž. Ja1,...,ans bgA: by˙˙˙a1yиии yans 0 and let J A be the set of all such JaŽ.ª. Note thatÄ4 0 s JŽ.0 g J ŽA .. Define a binary operation q on JŽ.Aby

JaŽ.Ž.Ž1,...,anqJb1,...,bmsJa1,...,an,b1,...,bm ..

Let JAŽ.s ²Ž.JA;q,0;Ä4::. For any commutative integral pomonoid B s ²:Ž.B; q,0;F , let F B be the set of nonempty order filters of ²:B; F . For F, G g F Ž.B , define F[Gto be the order filter of ²:B; F generated by Äf q g: f g F, ggG4Ä, i.e., F [ G s h g B: h G f q g for some f g F, g g G4, and define F y˙ G to be the filter of ²:B; F generated by DÄH g FŽ.B : H[G:F4. Let FBŽ.s ²F Ž.B;[,y˙,M;=:. THEOREM 2.1wx Pał82, OK85, Fle88 .Ž. i JA Ž . is a commutatië integral Ž. Ž . pomonoid, the map ␩A: a ¬ Ja agA is an isotone embedding of²:²Ž.: A; F into J A ; : and, for eëry a, b g A, we haëJaŽ.y˙b s FÄIgJŽ.A:IqJb Ž.=Ja Ž.4. Ž. Ž. Ä4Ž. ii FB is a pocrim and the map ␹ B: b ¬ c g B: b F cbgBisan isotone integral monoid embedding of B into²Ž. F B ; [ , M; = :which preserës existing residuals, i.e., wheneër a, b g B and c is the least element Ž. Ž. Ž. of B for which c q b G a, we haë ␹ BBBc s ␹ a y˙ ␹ b . Ž. Ä Ž. Ž. 4Ž. iii The map ␹JŽA. (␩A: a ¬ I g J A : Ja:IagA is a BCK- algebra embedding of A into the residuation reduct ŽFJA Ž Ž ...y of the pocrim FJAŽŽ ... 288 BLOK AND RAFTERY

Higgswx Hig84 observes that BCK-algebras may be defined as algebras of type² 2, 0 : satisfying Ž.Ž.Ž. 1 , 2 , 3 , and Ž. 5 , and proves that the quasivariety Mis not a variety. This extends Wronski’s´ resultwx Wro83 that BCK is not a variety. Let A be a pocrim or a BCK-algebra. By an ideal of A, we mean a subset I of A with 0 g I, such that a g I whenever a g A and a y˙b, b g I.An ideal of A is a hereditary subset of ²:ŽA; F . In the case that A is a pocrim, it follows quite easily that the ideals of A are just the nonempty subsets of A which are hereditary and closed under the monoid operation [..Ž. For any congruence or, indeed, tolerance ␪ of A, the 0-class 0r␪ s ÄagA:Ž.a,0 g␪4of ␪ is an ideal of A. Conversely, given an ideal I of A, there is at least one congruence whose 0-class is I, of which the largest, ÄŽ. 4 viz. ␸I[ a, b g A = A: a y˙˙b, b y a g I , is actually a relative congru- Ž. Ž. ence, i.e., Ar␸I abbreviated ArI satisfies the quasi-identity 5 , making n Ž.ArIyaBCK-algebra. A useful fact is that if ␶ g Tol A then 0r␶ s 0r␶ Ž. Ž. n for all positive n g ␻; consequently 0r␶ s 0r⌰ ␶ , since ⌰ ␶ s D ne␻␶ . These facts were proved inwx RRS91, Theorem 2.2 .Ž They depend only on Ž.2 and Ž.. 6 . The set of all ideals of A, ordered by inclusion, is an algebraic lattice which we denote by Id A;IdAis also distributivewx Pał81 . We denote by ²:XA the ideal of A generated by X : A, i.e., the intersection of all ideals of A containing X. Of course ²:л A s Ä40 and it is well known and not difficult to check that if л / X : A then

²: XAsÄ4agA:Ž.Ž᭚ng␻᭚x1,..., xngXa .Ž.y˙˙˙x1yиии yxns 0.

²: ² : Ä4 We abbreviate X AAby a1,...,amwhen X s a1,...,am. The maps I¬␸I and ␪ ¬ 0r␪ are isotone functions between the lattices Id A and Con A. Whereas I s 0r␸I for every ideal I, a congruence ␪ is generally ␸ A smaller than 0r␪ . It is well known that has the ideal extension property, i.e., whenever B g SAŽ.and I is an ideal of B, there is an ideal J of A such that J l B s I.Ž²:. We may simply take J s I A. We shall need the following simple result which provides a useful method of constructing pocrims, and hence also BCK-algebras.

LEMMA 2.2. Let A be a pocrim and let²: B; [ ,0 be a submonoid of ²:A;[,0 such that for e¨ery a g A, there is a least element b of ²: B; F for which a F b. Then the monoid²:²: B; [ ,0 is the [ ,0 -reduct of a pocrim B.

Proof. Let R be the residuation operation on A. For b12, b g B, define b12y˙ bto be the least element b of B such that b12R b F b. Clearly, b12y˙ bis also the least element b of B such that b12F b [ b . Thus, ²:B;[,0;F is residuated, with residuation operation y˙ , as required. VARIETIES OF RESIDUATED POMONOIDS 289

Note that we do not assume B to be closed under R, and the partial operation induced on B by R need not coincide with the restriction to its domain of the residuation operation y˙ on B.

3. PROPERTIES OF POCRIM VARIETIES

The following proposition is taken fromwx BR95 , where it was stated only in the context of BCK. The well-known arguments used there apply mutatis mutandis to the quasivariety M.

PROPOSITION 3.1. Let K be M or BCK and let A g K. Ž. Ž . Ž . i The maps ␪ ¬ 0r␪␪gCon K A and I ¬ ␸I I g Id A are mutually inërse lattice isomorphisms between the relatië congruence lattice of A and the ideal lattice of A. Ž.ii A is relatiëly 0-regular, is relatiëly congruence distributië, and has the relatië congruence extension property. Ž.iii HA Ž.:K iff A is 0-regular, in which case A is also congruence distributië. If HSŽ. A : K then A has the congruence extension property. In particular: Ž.iv Eëry subäriety of K is 0-regular and congruence distributië, with the congruence extension property.

The following easily checked factsŽ which are well known for BCK-algebras. go slightly further than the immediate consequences of the above proposition: a pocrim or BCK-algebra is subdirectly irreducibleŽ. resp. simple iff it has a smallestŽ. resp. a unique nonzero ideal. Thus, by Proposition 3.1, the notion of relative subdirect irreducibilityŽ. resp. relative simplicity with respect to M or BCK does not differ from its absolute counterpart.In particular, it follows from the relative congruence extension property that nontri¨ial subalgebras of simple pocrims or BCK-algebras are simple. Neither the quasivariety of pocrims nor the quasivariety of BCK algebras has the congruence extension property in the absolute sense, however wxBR93 . Any 0-regular variety is congruence n-permutable for some integer nG2wx Hag73 , so this is true of all pocrim varieties, byŽ. iv . The degree n of permutability turns out to be recognizable from the length of a scheme of identities, the satisfaction of which characterizes certain classes as pocrim varieties. This is the content of the next result.

THEOREM 3.2. Let K be a class of algebras of type ²:2, 2, 0 . Then HSPŽ.K is a ¨ariety of pocrims iff for some integer n ) 0, there exist 6-ary 290 BLOK AND RAFTERY

² : Ž.Ž.Ž.Ž. [,y˙ -terms t1,...,tn such that K satisfies 1, 2, 3, 4,and

x f tx1Ž.,y,xy˙˙y,yyx,0,0 ,Ž. 11 0

txiiŽ.,y,0,0, xy˙˙y, yyx ftxq1Ž.,y,xy ˙˙y,yyx,0,0

Ž.is1,...,ny1,Ž 11.i

txnŽ.,y,0,0, xy˙˙y, yyx fy.11Ž.n

In this case, HSPŽ.K is congruenceŽ. n q 1-permutable. Conërsely, if HSPŽ.K is a congruence Ž k q 1- .permutable äriety of pocrims then we may realise the aboë scheme of equations for some n F k. In other words, tnŽŽ..HSP K is the minimum integer n for which HSP Ž.K satisfies a scheme Ž. of equations of the form 110yn . Proof. Ž.« Certainly HSPŽK .satisfies Ž. 1 , Ž. 2 , Ž. 3 , and Ž. 4 . Let F s ²:F;[,y˙ , 0 be the HSP Ž.K -free algebra on two free generators x, y and F let J be the ideal of F generated by Ä4x y˙˙y, y y x . Let ␶ s TxŽŽ y˙ y,0.Ž , yy˙ x, 0 .. . Then by Proposition 3.1Ž.Ž. iv , since HSP K is a variety of pocrims, the relation ␸J is the unique congruence on F whose 0-class is J. F Ž. Since ␶ : ␸J and J is an ideal of F, we also have 0r␶ s J. Let ␪ s ⌰ ␶ . Ž. We know that 0r␪ s 0r␶ s J, so we have ␪ s ␸JJ. Now x, y g ␸ so n Ž.x,yg␶for some positive n g ␻, and if HSPŽ.ŽK is k q 1 . -permutable ŽŽŽ...i.e., tn HSP K F k , we may choose n F k. By Lemma 1.1, there exist polynomials G1,...,Gn on F with

x s Gx1Ž.y˙˙y,yyx,0,0 ,Ž. 12 0

Giiž/0,0, x y˙˙y, y y x s Gxq1Ž.y ˙˙y,yyx,0,0 Ž.is1,...,ny1,

Ž.12 i

Gnž/0,0, x y˙˙y, y y x s y.12Ž.n Ä4 ²: Now for each i g 1,...,n, there exist a [ , y˙ , 0 -term si and elements u ,...,u F with i1 imi g

GaŽ.,b,c,d suF ,...,u ,a,b,c,d Ž.13 iiis ž/1imi i FŽ. for all a, b, c, d g F. Find binary terms uijwith ux ij ,ysuij for each i, j and define

txŽ.Ž,y,z,z,z,z su x,y.Ž,...,ux,y.,z,z,z,z i1234siiŽ.1 imi 1234

Ž.Ž.is1,...,n .14i VARIETIES OF RESIDUATED POMONOIDS 291

Ž. Ž. Ž.F ŽHSP Ž.. It follows from 120yn0 , 13yn0 , and 14yn that and hence K Ž. satisfies 110yn1 . We may clearly assume that the terms t ,...,tnare free of occurrences of 0, in view of the identitiesŽ. 2 , Ž. 3 , and Ž. 6 and the monoid identities x [ 0 f x f 0 [ x. Now let ␶ be any tolerance on some A g HSPŽ.K . We show that n 1 n n1 n1 ␶q :␶. Let Ž.a, b g ␶ q. Then a y˙˙b, b y a g 0r␶ qs 0r␶ so A ␶ A tiiŽ.a,b,ay˙˙b,bya,0,0 tq1Ž.a,b,0,0,ay˙˙b,bya Ž.is1,...,ny1; Ž. Ž. ␶ n Ž.A A from 110yn it follows that a, b g , as required. Thus tn F n,so is congruence Ž.n q 1 -permutable. Ž . Ž. Ž. Ž. Ž. Ž .A HSP Ž . ¥ Let K satisfy 1 , 2 , 3 , 4 , and 110yn and let g K . Suppose a, b g A with a y˙˙b, b y a s 0. Since A satisfies all equations AŽ.AŽ. that hold in K, we have a s ta12,b,0,0,0,0 sta,b,0,0,0,0 s иии s AŽ. Ž. tan ,b,0,0,0,0 sb. This means that A satisfies 5 and is therefore a pocrim, as required. Every variety of pocrims known to us is congruence 3-permutable but we do not know whether this is true of all pocrim varieties. By contrast, it is known that every variety of BCK-algebras is congruence 3-permutable Ž.and no such nontrivial variety is congruence permutablewx Idz83 . An example showing that pocrim varieties need not be congruence permutable will be given in Section 6. If K is a variety of BCK-algebras with an equational base ⌺, it is easy to see that the class of all pocrims which are models of ⌺ is a variety V with SŽV y. : K. Here, V does not depend on the choice of ⌺. We need not have SŽV y.Žs K in general see Section 4, Example VI below. , but it turns out that SŽV y. is variety of BCK-algebras. This is a consequence of a stronger result, the next theorem, which characterizes those pocrim varieties V for which SŽV y. is a variety.

THEOREM 3.3.Ž. i Let V be a ¨ariety of pocrims. Then S ŽV y. is a ¨arietyŽ. of BCK-algebras iff there exist n, m g ␻ and binary ²y˙ :-terms ␣1,...,␣n,␤1,...,␤m such that V satisfies

x y˙˙˙˙˙˙␣1Ž.x, y y иии y␣n Ž.x, y f y y ␤1 Ž.x, y y иии y␤m Ž.Ž.x, y 15 Ž. Ž. and BCK satisfies the identities ␣ijx, x f 0 f ␤ x, x for 1 F i F n and 1FjFm.In this case, SŽV y.and V are congruence 3-permutable ¨arieties. Ž.ii If K is any class of algebras of type ²2, 2, 0 :satisfying the identities

Ž.1, Ž. 2, Ž. 3, and Ž.4,together with an identity of the form Ž15 .Žwith the ␣i Ž.. Ž . Ž and ␤j as in i then HSP K is a ¨ariety of pocrims and is congruence 3-permutable, by Ž..i. 292 BLOK AND RAFTERY

Proof. Ž.i Ž« .This follows from a result of Komori and Idziak Ž stated inwx Idz83. to the effect that every variety of BCK-algebras satisfies an identity of the formŽ. 15 , with the ␣ij, ␤ as described in the statement of the theorem, and is congruence 3-permutable.Ž Fuller proofs of these results may be found inwxwx BR95 or RS92 .. Ž.¥Let A be a subalgebra of By, where B g V . It suffices to show that HAŽ.:S ŽVy.. Let ␪ g Con A and let I s 0r␪. Let J be the ideal of B 2 Ž. 2 generated by I. Clearly ␪ : ␸JJl A . On the other hand, if a, b g ␸ l A AŽ. AŽ. AŽ. 22 then ␣iJia, b ␸␣ a,as0so␣ia,b g0r␸ JlA sJlA sIfor AŽ. Ž. is1,...,n. Similarly ␤j a, b g I for j s 1,...,m.By13, A A asay˙˙0yиии y ˙0 ␪ a y ˙␣1Ž.a, b y ˙иии y ˙␣n Ž.a, b A A sby˙˙˙˙˙˙␤1Ž.a,byиии y␤m Ž.a, b ␪ b y 0 y иии y 0 s b. 2 ŽŽ .y. Thus, ␪ s ␸JJl A , hence Ar␪ g SBr␸ , where Br␸Jg V . Ž.ii Certainly HSPŽ.K is a variety of type² 2, 2, 0 : satisfying all identities that hold in all members of K. For 1 F i F n and 1 F j F m, in view Ž. Ž. of the fact that BCK satisfies ␣ijx, x f 0 f ␤ x, x , it follows from Ž. Ž . Ž . wxBR95, Lemma 7 that BCK satisfies ␣iix, y y˙˙˙˙pxyyyqy iyxf Ž. Ž . Ž . 0f␤jjx,yy˙˙˙˙kxyyyly jyxfor some integers pi, qi, k j, l jg ␻.If we define

Ž. tx1 ,y,z,u,¨,w Ž. sxy˙˙˙˙˙˙˙␣111Ž.x,yypzyqu yиии yw␣nnnx, y y pzyqux, Ž. tx2 ,y,z,u,¨,w Ž. syy˙˙˙˙˙˙˙␤111Ž.x,yyk¨ylw yиии yw␤mmmx, y y k ¨ y lwx, then equationsŽ.Ž. 1101 , 11 , and Ž. 11 2 of Theorem 3.2 hold, byŽ. 15 and Ž. 2 , and the result follows from Theorem 3.2. The next result offers a sufficient condition for a variety of pocrims to be congruence permutable. All congruence permutable pocrim varieties known to us satisfy this conditionŽ the examples in the next section and Proposition 6.7 show that it is a useful test for permutability. , but we do not know whether it characterizes congruence permutability for pocrim varieties. ²: THEOREM 3.4. Let k g ␻ and let u1,...,un,w1,...,wm be binary y˙ - Ž. Ž. terms such that BCK satisfies uij x, x f 0 f wx,x for 1 F i F n and 1FjFm.If A is a pocrim satisfying

y y˙˙˙ux1Ž.,yyиии yuxn Ž.,y[wx1 Ž.,y[иии [ wxm Ž.,y fx[kyŽ.y˙x Ž.16 VARIETIES OF RESIDUATED POMONOIDS 293 then HSPŽ. A is a congruence permutable Ž hence arithmetical .¨ariety of pocrims. Proof. An argument similar to that of Theorem 3.3Ž. ii shows that the variety HSPŽ. A consists of pocrims, hence this variety is congruence distributive, by Proposition 3.1. To prove congruence permutability, define

txŽ,y,z .szy˙˙˙ Ž.yyxyux1Ž.,yy ˙˙ Ž.yyx

y˙˙иии yuxnŽ.,yy ˙˙Ž.yyx

[wx1Ž.Ž.,yy˙˙Ž.yyx [иии [ wxm ,yy˙˙Ž.yyx. Ž. Ž. Ž. Ž. Then A satisfies tx,x,yfwyy˙˙˙ux1,xyиии yuxn,xx[wx1,x Ž . Ž.Ž. . [иии [ wxm ,xfyby 2 , 6 , and y [ 0 f y , as well as

txŽ,y,y .f yy˙˙˙ Ž.yyxyux1Ž.,yy ˙˙ Ž.yyx

y˙˙иии yuxnŽ.,yy ˙˙Ž.yyx

[wx1Ž.Ž.,yy˙˙Ž.yyx [иии [ wxm ,yy˙˙Ž.yyx fx[kyŽ.Ž.y˙˙Ž.yyxy ˙xfx[k0fx

ŽŽ.by 16 and the fact that y y˙˙Ž.y y x is always a lower bound of x in any pocrim. .

4. EXAMPLES OF POCRIM VARIETIES

Many well-known varieties from classical algebra and from logic consist of pocrims. We give examples of these and other pocrim varieties.

EXAMPLE IŽ. The quasivariety generated by the natural numbers . We have already observed that the set ␻ of all nonnegative integers is the universe of a pocrim ␻ s ²:␻; q, y˙ , 0 , where addition and the partial order are natural. It is known that the quasivariety ISPPuŽ.␻ is a variety wxBF93, Corollary 2.4 which is equationally completew Ame84x and therefore an atom of P¨ Ž.M . For reasons to be discussed in the examples that follow and in Section 8, the members of this variety are called cancellati¨e hoops inwx BF93 , the results of which imply that an equational base for this variety is provided by the identitiesŽ.Ž.Ž. 1 , 2 , 3 , and Ž. 4 together with the two following identities:

x [ Ž.y y˙˙x f y [ Ž.x y y ,17Ž. Ž.x[yy˙yfx.18Ž. 294 BLOK AND RAFTERY

Let ²:L; q, y,0,k,n be a lattice ordered Abelian group, i.e., ²:L;q,y, 0 is an Abelian group and there is a lattice order F on L, compatible with q, whose supremum and infimum operations are k and n, respectively. For a, b g L, define a y˙ b s Ž.a y b k 0. If Lqis the positive cone Ä4a g L:0Fa of L then Lqs ²Lq; q, y˙ , 0: is a pocrim satisfyingŽ. 17 and Ž. 18 , hence a member of ISPPuŽ.␻ ; conversely, any member of ISSPuŽ.␻ is obtainable from the positive cone of a lattice ordered Abelian group in this way. This correspondence between ISPPuŽ.␻ and lattice ordered Abelian groups arises in G. Birkhoff’s work on ‘‘Abelian l-groups’’wx Bir84 and is also stated in w Bos69 xwx , Ame84 , and w Pon86 x . Moreover, it may be described as an equivalence of categoriesw Fer92, Theorem 4.7x . The variety ISPPuŽ.␻ also satisfies

x y˙˙Ž.x y y f y y ˙˙ Ž.y y x , Ž.T a special case ofŽ. 15 , thus illustrating Theorem 3.3.

EXAMPLE IIŽ.Ž. The variety of Brouwerian semilattices . A dual Brou- werian semilattice Žseewxwx Koh81¨˙ and BP94.²: is an algebra A; k , y,0 of type²: 2, 2, 0 , where ²A; k :is a join semilattice with least element 0, and for all a, b g A, the condition

᭙ xaŽ.Fbkxlay˙bFx is satisfied. It follows immediately that a Brouwerian semilattice is a pocrim, with monoid operation k, satisfying the idempotent identity

x [ x f x. Ž.M1

Conversely, any pocrim satisfying Ž.M1 has the property that its underlying poset is an upper semilattice whose supremum operation coincides with the monoid operation.ŽŽ.Ž.. This follows quite easily from M1 and 10 . We therefore denote the class of Brouwerian semilattices by M1; it is a variety. Ž y.² : It is known that the class S M11of y˙ , 0 -subreducts of M is just the Ž variety E1 of Hilbert algebras wxDie66 also known as positi¨e implicati¨e BCK-algebras Že.g.,wx Cor82a.. . The members of E1 may be characterized as BCK-algebras satisfying the identity

Ž.x y˙˙y y y f x y ˙y. Ž.E1 This is equivalent, over BCK, to the identity

x y˙˙˙˙Ž.Ž.x y y y y y x f y y ˙˙˙˙ Ž.Ž.x y y y y y x . Ž.E1Ј

This fact illustrates Theorem 3.3, since Ž.E1 Ј is clearly a special case of Ž.15 . VARIETIES OF RESIDUATED POMONOIDS 295

¨ The lattice L Ž.M1 of varieties of Brouwerian semilattices has a unique atom, viz., the variety of generalized Boolean algebras, which we denote by G. Recall that a generalized Boolean lattice is just a distributive lattice ²:L;k,nwith a least element 0, which is relati¨ely complemented in the sense that whenever a, b g L with b - a, the quotient sublattice arb [ Ä4cgL:bFcFais complemented. If we define a y˙ b to be the comple- ment of b in Ž.a k b r0 then the algebra ²L; k , y˙ , 0 : is called a generalized Boolean algebra and is a Brouwerian semilattice. The variety G of all generalized Boolean algebras may be characterized as the class of all pocrims satisfying

x y˙˙Ž.y y x f x.19Ž. Ž. Alternatively, G s ISP C22, where C is the unique two-element pocrim, i.e., the pocrim reduct of the two-element Boolean algebra. Also, G and ¨ ISPPuŽ.␻ are the only atoms known to us in the poset P Ž.M of varieties of pocrims. The class SŽ Gy.²of y˙ , 0 : -subreducts of members of G is the variety of Tarski algebras wxMon60Ž alias implication algebras wxMit71 or implicati¨e BCK-algebras wxCor82a.Ž. ; this is the unique smallest nontrivial variety of BCK-algebras and is axiomatized byŽ.Ž.Ž. 1 , 2 , 3 , and Ž 19 . . Tarski algebras also satisfy Ž.T . The variety M of Brouwerian semilattices is known to have 2 / 0 subvari- etieswx Koh81,¨ Owe74 . Thus, there are at least 2/0 varieties of pocrims and since pocrims have finite type, it follows that

Ž.᭙a, b g AbFal Ž.Ž.᭚cgAasbqc. These algebras were first investigated in a manuscript of J. R. Buchi¨ and T. M. Owens, entitled ‘‘Complemented Monoids and Hoops,’’ circa 1975. For more recent and comprehensive work on these algebras, seewx Ame82 , wxwxwxwxwxAme84 , BP94 , BF93 , BF and Fer92 . It is knownŽ e.g.,w BP94, Lemma 1.3x.² that if A is a hoop then the underlying poset A; F :is an upper semilattice whose join operation may be defined by x k y s x [ Ž.y y˙ x . It follows easily that hoops are just those pocrims which satisfy the identity Ž.17 , and consequently that the class of all hoops Ž which we denote by H . is a variety for whichŽ.Ž.Ž.Ž. 1 , 2 , 3 , 4 , and Ž 17 . serve as an equational base. It is known that the lattice of subvarieties of H has exactly two atoms, viz.,

Gand ISPPuŽ.␻ wxAme84 , and that the variety M1 of Brouwerian semilattices consists of hoops. Thus, all examples of pocrim ¨arieties mentioned thus far are ¨arieties of hoops. We have remarked that the ideal lattice of a commutative ring with 1 is a pocrimŽ with respect to ideal multiplication 296 BLOK AND RAFTERY and reversed set inclusion. ; in particular, the ideal lattice of a Dedekind domain is aŽ. cancellative hoop, i.e., a member of ISPPu Ž.␻ . Hoops are congruence permutableŽ. and therefore arithmetical, by Proposition 3.1 since the term txŽ,y,z .s Žxy˙˙ Žyyz ..k Žzy ˙˙ Žyyx .. is a Mal’cev term, i.e., H satisfies txŽ.,x,yfyand tx Ž.,y,yfx. Notice also thatŽ. 17 is of the form of the identityŽ. 16 of Theorem 3.4Ž take n s 0; m s k s 1 and Ž. . wx1 ,ysxy˙ y, giving an alternative proof that H is congruence permutable. Not every congruence permutable variety of pocrims consists of hoopsŽ. see the remarks following Theorem 6.1 . It is also known that hoops satisfy a useful identity introduced by Cornishwx Cor81 , viz.,

x y˙˙˙˙Ž.x yŽ.y yŽ.y y x f y y ˙˙˙˙Ž.y y Ž.x yŽ.x y yJ Ž. wxBF93, Corollary 4.8 . Thus, by Theorem 3.3Ž. i , the class S ŽV y.is a variety of BCK-algebras for every subvariety V of H Žin particular, for all of the aforementioned pocrim varieties.Ž . The variety S H y.is just the class of all BCK-algebras satisfying

Ž.Ž.Ž.Ž.z y˙˙˙x y y y x f z y ˙˙˙y y x y yH Ž. wxBF93, Corollary 4.5 . An equational base for this variety is therefore given byŽ. 1, Ž. 2, Ž. 3, ŽH ., and ŽJ .. The residuation reducts of hoops have also been studied by Cornishwx Cor82b, Cor84 . Notice that the identity Ž.J is a consequence of the identity Ž.ŽT over BCKand hence over M .Ž. Pocrims or BCK-algebras satisfying T .are called commutatië. Such algebras were first studied by Tanakawx Tan75 , who showed that their underlying posets are lower semilattices in which the infimum operation is definable by x n y s x y˙˙Ž.x y y . We call a BCK-algebra A directed Ž.²:resp. bounded if the poset A; F is upward directedŽ. resp. has a greatest element . Every directed commutative BCK- algebra satisfies Ž.Ž.x y˙˙y n y y x f 0wx Tra79, Pał80 . This is equivalent, over BCK, to the identity z y˙˙˙ŽŽz y x y y ..ŽF z y ˙˙y y x ., the satisfaction of which characterizes a BCK-algebra as a subdirect product of ones with linearly ordered underlying posetswx Pał80 . Thus in particular, eëry subdirectly irreducible commutatië pocrim is linearly ordered. It is well known that every directed commutative BCK-algebra is embeddable into a bounded commutative BCK-algebra. More strongly, any directed BCK-algebra is embeddable into an ultraproduct of its bounded subalgebras: this isw BF93, Lemma 4.1xw and was also discovered independently by T. Sturm Stux . Applying some of these results we obtain further examples of hoops:

PROPOSITION 4.1. Eëry commutatië pocrim is a hoop. Thus, the äriety of commutatië pocrims is a subäriety of H. VARIETIES OF RESIDUATED POMONOIDS 297

Proof. Let A be a commutative pocrim. Since Ay is directed, it is a subalgebra of a bounded commutative BCK-algebra ²:B; y˙ , 0 with greatest element 1, say. If we define [ on B by a [ b s 1 y˙˙˙Ž.1 y a y b Ž.²a,bgBthen B; [ , y˙ , 0 : is a pocrim and, in fact, a hoop.Ž More strongly, if we define a s 1 y˙˙a and a и b s a y b then ²:B; [ , и , , 0, 1 is an MV-algebra in the sense of Changwxw Cha58 : see Mun86 x .. Thus, AygSŽHy.Žand so A satisfies H .Ž. It follows from 4. that A satisfies zy˙˙˙˙w x[Ž.yyxxwfzyy[Ž.xyyx. Replacing z by x [ Ž.y y˙x ,we find that A satisfies 0 f wx [ Ž.y y˙˙x xwy y [ Ž.x y ˙y xand, by symmetry, 0fwy[Ž.xy˙˙yxwyx[Ž.yy ˙xx.By5,Ž.Asatisfies Ž 17 . , i.e., A is a hoop. This means that commutative pocrims coincide with ‘‘Wajsberg hoops’’ in the sense ofwx BP94 , i.e, with the hoop subreducts of MV-algebras. We shall now consider some pocrim varieties not consisting entirely of hoops.

EXAMPLE IVŽ. The varieties of n-potent pocrims . Given n g ␻,a pocrim is said to be Ž.n q 1-potent if it satisfies any one of the following equivalent identities, whose equivalence is proved inw BR95, Proposition 13 and Lemma 14x :

Ž.n q 1 x f nx; Ž.Mn

xy˙˙Ž.nq1yfxyny, Ž.En

xy˙˙˙˙nxŽ.Ž.yyynyyxfyy ˙˙˙˙nx Ž.Ž.Ž.yyynyyx. EnЈ

Note that for any integers i, j, m, n g ␻, the identity

ij x y˙˙˙˙ixŽ.Ž.yyyjyyxfyy ˙mx Ž.Ž.y ˙˙˙yynyyxCŽ.mn is of the form ofŽ. 15 . Since ŽEn .Ј is a special case of this, it follows from Ž. Theorem 3.3 that for each n g ␻, the class Mn of all n q 1 -potent pocrims is a congruence 3-permutable variety for which the identitiesŽ. 1 ,

Ž.2 , Ž. 3 , Ž. 4 , and any one of ŽMnn ., ŽE ., and ŽE n .Ј serve as an equational y base, and that the class SŽMn .Ž.is also a variety of BCK-algebras . The variety of BCK-algebras axiomatized by the identitiesŽ.Ž.Ž. 1 , 2 , 3 , and ŽEn . ŽŽor equivalently Enn.Ј.is usually denoted by E . Of course, M11and E are the aforementioned varieties of Brouwerian semilattices and of Hilbert Ž y. algebras; we have mentioned that S Mnn: E ; whether or not equality holds for n ) 1 is an open problem. We have also mentioned that Brouwerian semilattices are congruence permutable but we shall show in

Section 6 that for each n ) 1, the variety Mn is not congruence per- Ž. mutable hence Mn H . It is known and quite easy to see that every finite BCK-algebraŽ resp. .Ž. pocrim satisfies En for some n g ␻ and therefore generates a variety of 298 BLOK AND RAFTERY

BCK-algebrasŽ. resp. pocrims . Also, if A is a subdirectly irreducible algebra ²: in any of the varieties En then the poset A; F has a unique atom wxPała . Of course, E0 is the trivial variety; the varieties Ennand M are locally finite iff n F 1wx Die66, Dyr87 . The chains of varieties Mn, n g ␻, and En, n g ␻, are strictly increasingwx Cor82a . It is easy to see that the equationally complete variety ISPPuŽ.␻ meets each of the varieties Mn trivially. Recall that a variety V is said to have definable principal congruences Ž.DPC if there is a 4-ary first-order formula ␸ in the language of V such that for all A g V and all a, b, c, d g A, A c, d g ⌰ Ž.a, b m A * ␸ Ža, b, c, d .. If, moreover, ␸ may be chosen to be aŽ. finite conjunction of equations then we say that V has equationally definable principal congruences Ž.EDPC . All varieties with a ternary deductive term have EDPCw BP94, Corollary 2.5x . Here, a ternary deducti¨e term Ž.briefly, a TD term for a variety V means a ternary term p of the language of V such that V satisfies A pxŽ.,x,zfzand for all A g V and a, b, c, d g A,if Ž.c,d g⌰ Ž.a,b A A then paŽ.,b,cspaŽ.,b,d. Such a term p is called commutati¨e if V also satisfies pxŽ,y,px ŽЈ,yЈ,z ..fpx ŽЈ,yЈ,px Ž,y,z ..; it is called regular if for some constant term e of V and all A g V , we have Ž.a, b g A Ž A Ž A . A . ⌰ pa,b, e , e whenever a, b g A. The varieties Mnnand E have EDPC; in fact, the term pxŽ.,y,zszy˙˙˙˙nx Ž.Ž.yyynyyx is a commu- tativeŽ. but not regular TD term for both of themwx Pałb and it is not hard to showŽ. e.g., using Lemma 4.3 below that qxŽ,y,z.sz[nxŽy˙y.[ Ž. nyy˙ x is a commutative regular TD term for Mn, with respect to the constant 0. The following is a converse of these results:

THEOREM 4.2. The following conditions on a äriety V of pocrimsŽ resp. of BCK-algebras. are equiälent: 1. V has EDPC; 2. V has DPC; Ž. 3. V is a subäriety of Mnnresp. of E for some n g ␻. Proof. Statements 1 and 2 are equivalent for any congruence distributive variety with the congruence extension propertywx BP88, Corollary 4.8 , so their equivalence in the present context follows from Proposition 3.1. It remains only to prove that statement 2 implies statement 3. Suppose V has DPC. In view of the isomorphism between the congruence and ideal lattices of members of V Ž.Proposition 3.1 , this implies the existence of a binary first-order formula ␸ in the language of V such that for all A g V and all a, b g A,

ag²:bAm A*␸Ž.a,b. VARIETIES OF RESIDUATED POMONOIDS 299

By the characterization of ideal generation given earlier, this clearly means that

A * ␸ Ž.a, b m Ž᭚n g ␻ .Ža y˙nb s 0. .

We claim, however, that there is a fixed n g ␻ such that

ag²:bA m ay˙ nb s 0.

For if not, then for every n g ␻, there exist A nng V and a , bng A such ²: that anng b A nbut a nny˙ nb / 0. Let U be a free ultrafilter over ␻, let A Ł A Ž. be the ultraproduct ng␻ nrU and consider a s a01, a ,... , bs Ž.⌸ A ␸Ž. ␻ b01,b,... g ng␻Annn. Since * a , bnfor each n g , we have that A * ␸Ž.arU, brU and so arU g ²:brU A. But for each n g ␻, Äig␻:aiŽ.y˙˙nb Ž. i / 04Ä= i g ␻: i G n 4g U,soarUynbrU / 0, contradicting arU g ²:brU A. This vindicates the above claim. If n is as in Ž. the claim, then V : Mnnresp. E . For otherwise, we may choose B g V and a, b g B such that a y˙˙nb / a yŽ.n q 1 b, whence

ay˙˙Ž.ayŽ.nq1by ˙nb s Ža y ˙nb .y ˙˙ Ž.a y Ž.n q 1 b / 0.

Thus, if c s a y˙˙ŽŽa y n q 1 ..b then c f ²:b B. But this contradicts the fact that c y˙ Ž.n q 1 b s 0, completing the proof. Ž. LEMMA 4.3. Let A be a BCK-algebra in En n g ␻ and let I be the ideal of A generated by b1,...,bm gA. Ž. Ä4 iIsagA:ay˙˙˙nb1y иии ynbms 0. In particular, if A is the ²: y Ž.Ä y˙ ,0 -reduct B of a pocrim B e. g., if B g Mn then I s a g B: ay˙ nb s 04Äs a g B: a F nb4 is the principal ideal of B generated by b s b1 [ иии [ bm. Thus, e¨ery finitely generated congruence of an algebra in Mn is principal. Ž. Ž. ii The map f: A ª A defined by f a s a y˙˙˙nb1y иии ynbm is a homomorphism of BCK-algebras and 0rker f s I. Proof. Ž.i This follows from the characterization of ideals generated by subsets given in Section 2 and the identities ŽEn .Ž., 4 and Ž. 8 . Ž. ii Let a, c g A. Abbreviating a y˙˙˙nb1y иии ynbmiias a y ˙Ý nb ,we have c G c y˙ Ýiinb , hence

Ž.ay˙˙cyÝÝnbiis a y ˙nb y ˙c Ž.byŽ. 8 ii

anb c nb . Fž/ž/y˙˙˙ÝÝiiy y ii 300 BLOK AND RAFTERY

Conversely,

a nb c nb ž/ž/y˙˙˙ÝÝiiy y ii

anb nb c nb by Ž.E and Ž. 8 sy˙˙˙˙ÝÝiiy yž/y Ý iŽ. n ii i

anb c nb nb Ž.byŽ. 8 sy˙˙˙˙ÝÝÝiiiyž/y y iii

FŽ.ay˙˙cyÝnbi i

by isotonicity of x nb and repeated application ofŽ. 9 , ž/y˙Ý i i as required. ByŽ. i and Proposition 3.1, 0rker f s I.

THEOREM 4.4. For each n g ␻, eëry subquasiäriety of En is a äriety.

Proof. Let Q be a subquasivariety of En. We have to show that HŽ.Q:Q. Let A g Q and let g: A ª B be a surjective homomorphism. Suppose B f Q. Then there is a quasi-identity

ªª ªª &sxiiŽ.ftx Ž.ª sx Ž.ftx Ž. Ž.␣ is1 in the language of BCK-algebras such that Q * Ž.␣ and B ^ Ž.␣ . Choose ªªŽ . BBŽ. Ž.ª cabbreviating c1,...,cniigB such that scstcfor i s 1,...,k, BBªªª but scŽ./tc Ž.. Let a abbreviate a sequence of preimages a1,...,an of AAŽŽŽ.ªª A Ž.. ŽAA Ž. ªª Ž... c1,...,cn under g. Set ␪ s ⌰ sa11,ta,..., sakk,ta. Then AA ␪:ker g and ŽŽ.Ž..saªª,taf␪,soAr␪^Ž.␣ . Since A belongs to theŽ. 0-regular subvariety En of BCK, it follows that Ž. ␪s␸I , where I s 0r␪ g Id A Proposition 3.1 . Thus, Ar␪ s ArI, where ²: Iis finitely generated, say I s b1,...,bm A. By the previous lemma, the Ž. map f: A ª A defined by fdsdy˙˙˙nb1y иии ynbmis a homomorphism and 0rker f s I, so ker f s ␪, by 0-regularity. By the homomorphism theorem, Ar␪ s Arker f ( f wxA g SAŽ.:Q,soAr␪* Ž␣ ., a contradiction. Our argument does not extend to a proof that every subquasivariety of

Mn is a variety, since it is easy to find examples of algebras in M2 for which maps f as in Lemma 4.3Ž. ii are not [-homomorphisms. In fact, a subquasivariety of M3 which is not a variety may be constructed as follows. Let AsÄŽ.n,ig␻=␻:nF2, i F 14 and define a linear order on A by VARIETIES OF RESIDUATED POMONOIDS 301

Ž.Žn,i-m,j .iff either n - m or both n s m and i - j. Define a binary operation on A as follows, where x, y g A and a g Ä40, 1, 2 : x [ y s y [ x, x[Ž.0, 0 s x,0,1 Ž.Ž.Ž.Ž.Ž.Ž.Ž.[a,1 s a,1 s 0, 1 [ a,0, 1,0 [ 1, 0 s Ž.2, 0 , and x [ y s Ž.2, 1 in all remaining cases. Then ²A; [ ,0,0; Ž.F :is a commutative integral pomonoid which must be residuated, since it is well ordered. Let A be the resultant pocrim, which is clearly in M3. Note that A has an ideal I s ÄŽ.Ž.0, 0 , 0, 14 such that ArI is isomorphic to the unique Ä4Ž. pocrim B3 on 0, 1, 2 ordered naturally in which 1 [ 1 s 2. Let Q : M3 Ž. be the quasivariety generated by A. Since A is finite, QSI : IS A . But B3 is simple and it is easily checked that B3 is not isomorphic to any subalgebra of A,so Qis not a variety.

Our investigation of the varieties Mnnand E will continue in Sections 6 and 7. Here we give further examples of pocrim varieties which are not contained in the variety of hoops, nor in any of the varieties Mn. i j EXAMPLE V. For any i, j, m, n g ␻, the class Kmn of all pocrims satisfying

ij x [ ixŽ.Ž.y˙˙y[jyyxfy[mx Ž.Ž.y ˙y[nyy ˙xKŽ.mn is a congruence 3-permutable variety. This follows by setting n s 2, Ž. Ž . tx12,y,z,u,¨,wsx[i¨[jw, and tx,y,z,u,¨,wsy[mz [ nu in Theorem 3.2. It would be interesting to know whether all of these varieties 12 satisfy the conditions of Theorem 3.3Ž. i . It can be shown that K21 is not congruence permutable.

EXAMPLE VI. For each n gyÄ41j␻, we define, inductively, a ²: Ž. y˙ , 0 -term jxn ,yby

jxy1Ž.,ysx,

jx2nŽ.,ysyy˙˙Ž.yyŽ.jx2ny1Ž.,y,

jx2nq12Ž.,ysxy˙˙Ž.xyŽ.jxnŽ.,y, and an identity:

jxnnŽ.,yfjy Ž.,x. Ž.Jn

These identities were introduced inwx BR95 . As an easily application of Theorem 3.3, the class Jnnof all pocrims satisfying Ž.J is a Ž congruence 3-permutable. variety for each n g ␻. Moreover, it follows fromw BR95, Lemma 17x that every finite pocrim lies in some Jn and that the sequence of varieties Jn, n g ␻, is an ascending chain. We shall need these varieties in Section 8, where we shall show in particular that this chain is strictly 302 BLOK AND RAFTERY ascendingŽ. Corollary 8.11 . The reader should be warned that inwx BR95 , Jn denotes the variety of BCK-algebrasŽ. rather than pocrims satisfying Ž.Jn. The identity Ž.J0 is just the identity Ž.T defining commutative pocrims and BCK-algebras. We have mentioned that subdirectly irreducible commutative pocrims are linearly ordered. Since there exist simple commutative BCK-algebras which are not linearly orderedŽ e.g.,w Stu82, Example y 2ex.Ž , it follows that S J0 .is properly contained in the variety of all commutative BCK-algebras. Proposition 4.1 says that J0 : H. Note that Ž.J1 is just Cornish’s identity Ž.J , mentioned in Example III above, and so Ž. Ž. H:J1. Since the pocrim ␻ satisfies T , we have ISPPu ␻ : J0. The last three inclusions are all sharp, as can be deduced by considering, respectively, the example inwx Tra88, 3, Remark 3 , the four-element pocrim A to be discussed in the sequel to Theorem 6.1 and the two-element pocrim C2 .

5. ORDINAL SUMS

A natural technique for generating pocrimsŽ. and hence BCK-algebras is A the ‘‘ordinal sum’’ construction. Given pocrims A s ² A; [ , y˙ , 0: and B A B s ² B; [ , y˙ , 0: with A l B : Ä04 , we define the ordinal sum A q B B of A and B to be the pocrim with universe A q B s A j Ž B R Ä04. , the restrictions of whose partial order F Ž.resp. monoid operation [ to A B and to B R Ä04 coincide with the original ordersŽ resp. monoid operations.Ž on A and B, having the additional property that a - b resp. B A a [ b s b. whenever a g A, b g B R Ä04 . Consequently 0 is the least element of A q B and the restriction to A of the residuation operation y˙ on A q B coincides with residuation in A. For a g A and b, bЈ g B R B A Ä04 , we have a y˙˙b s 0 and b y a s b, while b y ˙bЈ is as in B unless A bFbЈ, in which case, b y˙ bЈ s 0 . A If A l B Ä04 , we abuse notation by writing A q B for the ordinal A sum A q BЈ, where BЈ is any algebra with BЈ ( B and A l BЈ : Ä0.4 If A and B are BCK-algebras, we define the ordinal sum A q B similarly, omitting the specification of [ and using the above characterization of residuation as the definition of y˙ . It is well known that A q B g BCK in this case. Moreover, for pocrims A and B, we have Ž.A q B yyys A q B . If A and B are both pocrims or both BCK-algebras, then the ideals of A q B are just the ideals of A, together with all sets of the form A j Ž J R Ä0B4. , where J is an ideal of B. Note also that both A and B are isomorphic to subalgebras of A q B, and that Ž.A q B rA ( B. A variety V of pocrims or of BCK-algebras is said to be closed under A ordinal sums if A q B g V whenever A, B g V and A l B : Ä04 . The following proposition is easy to prove and shows that closure under ordinal sums is a very common property. VARIETIES OF RESIDUATED POMONOIDS 303

PROPOSITION 5.1. The following ¨arieties are closed under ordinal sums: Ž.␻mŽnq1.Ž.Ž.␻ ␻ H,Mnng,KŽmq1.nnm,ng,Eng,

i j ½5AgMŽ.resp. BCK : A * Ž.Cmn Ž1 F i, j, m, n g ␻ .,

Ä4A g M Ž.Ž.Ž.resp. BCK : A * Jn1 F n g ␻ , and any ¨ariety of the form SŽV y., where V is a ¨ariety of pocrims that is closed under ordinal sums.

On the other hand, G, ISPPuŽ.␻ , J0 , and the variety of commutative BCK-algebras are not closed under ordinal sums. The wide applicability of this closure property leads us to consider the following recursive construction. For the rest of this section, W shall denote any ¨ariety of pocrims or of BCK-algebras that is closed under ordinal sums and V shall denote an arbitrary sub¨ariety of W . ii i We define classes K s K Ž.ŽV i g ␻ .and varieties V and V q Žig␻jyÄ41. as follows:

y1 0 0 V is the trivial subvariety of V ; K s V SI , V s V ;

m my1 m m K s Ä4A q B: A g V SI and B g V , V s HSPŽ.K , 0 - m g ␻ ; k V q is the supremum in L¨ Ž.W of the set Ä V : k g ␻4 , i.e., V q

k sHSPž/D V . kg␻

i 1 i Of course, V y : V : V q: W for all i g ␻. The only role that W will play is that its existence ensures that V q consists of pocrims or BCK-algebras and is therefore congruence distributive; in particular, Jonsson’s´ theorem is applicable to V q. Actually, every candidate for V known to us is contained in some variety of pocrims or BCK-algebras that is closed under ordinal sums. Consequently, although we have not proved the existence of W to be a universally satisfied requirement, we regard it as a weak assumption. Ä LEMMA 5.2. Let V Ј be a sub¨ariety of W with V : V Ј and let A i: 4Ä 4 igI and Bi: i g I be families of algebras in V and V Ј, respecti¨ely. Then Ä4 Ä any C g PAu iiqB:igI is an ordinal sum A q B, with A g PAui: igI4:Vand B g V Ј. Proof. Assume first that W : M. Consider the enlargement ² [ , y˙ ,0; P: of the language of pocrims by a unary predicate symbol P. Let ⌺ 304 BLOK AND RAFTERY be a set of identities axiomatizing V Ј. Let ⌫ be the union of ⌺ and the set consisting of the following first-order sentences Ž x F y abbreviates x y˙ y f 0:.

᭙x᭙yPxŽ.Ž.&Py Ž.ªPx Ž[y . ᭙x᭙yPxŽ.Ž.&Py Ž.ªPx Žy˙y . PŽ.0

᭙x᭙y Ž.Ž.!PxŽ.&!Py Ž.ª!Px Ž[y . ᭙x᭙yŽ.Ž.Ž.!PxŽ.&!Py Ž.&xFuyª!PxŽ.y˙y ᭙x᭙yPxŽ.Ž.&!Py Ž.ª ŽxFy&yy˙xfy. ᭙x᭙yPxŽ.Ž.&!Py Ž.ªx[yfy.

Let C iiis A q B for each i g I. Each C ibecomes a model of ⌫ if we C i C Ł C interpret P as Aii.If J:Iand s gJirU, where U is an ultrafil- C ter over J then C * ⌫. If we define A s Äa g C: PaŽ.4and B s Ž.C R A C j Ä04 then A and B are universes of subalgebras A, B of C g V Ј and C A B Ł Ä Ž. s q .If crUgC, where c g igJiC then either K s i g J: ci 4 A Ł A gAiior J R K is an element of U,so s gJirUgV. If W : BCK, we delete all sentences involving [ from ⌫ and use the same argument.

PROPOSITION 5.3. For each m g ␻, the subdirectly irreducible algebras in m the ¨ariety V are just those algebras isomorphic to ordinal sums A q B, my1 where A g V SI and B g V .

Proof. Clearly any ordinal sum A q B with A g V SI is subdirectly irreducible; its least nonzero ideal coincides with that of A. The converse is proved by induction on m and is clearly true when m s 0. Let m ) 0. Recall that W is congruence distributiveŽ. Proposition 3.1 so by Jonsson’s´ m Ž m.Žm. theorem, V SI : HSPuuK . By the previous lemma, P K consists of m 1 ordinal sums A q B with A g V and B g V y , where A is nontrivial. my1 m Since V and V are varieties, every member of SPuŽ K . is either in m 1 m 1 V y or is also an ordinal sum A q B with A g V , B g V y ŽA m nontrivial.Ž . Likewise, the same is true of members of HSPu K .. Thus, if mmy1 Ž CgVSI then either C g V SI or C s D q E with D g V hence . my1 D g V SI and E g V . In the latter case, C has the desired form. The same conclusion follows in the former case, by the induction hypothesis. Recall that a variety is semisimple if all of its subdirectly irreducible members are simple. VARIETIES OF RESIDUATED POMONOIDS 305

COROLLARY 5.4. If V is a semisimple ¨ariety then for each m g ␻,

m my1 V SI s Ä4A q B: A g V is simple and B g V .

PROPOSITION 5.5. For each r g ␻, e¨ery r-generated algebra in V q is a member of V ry1. Proof. The proof is by induction on r. The result is clearly true when r s 0, so assume r ) 0 and consider a subdirectly irreducible r-generated m algebra C g V , where m g ␻. By Proposition 5.3, C s A q B, where my1 Ä A 4 A g V SI and B g V . Since A is nontrivial and B j 0 is a subuni- A verse of C, at least one generator of C lies in A R Ä04 , so the algebra r2 B(CrAis Ž.r y 1 -generated. By the induction hypothesis, B g V yso r 1 C g V y . Now the V q-free algebra on r free generators is a subdirect k product of subdirectly irreducible algebras in D k g ␻ V , each on at most r generators, and this completes the proof.

COROLLARY 5.6. For any äriety V of pocrimsŽ. resp. BCK-algebras , if there is a äriety U of pocrimsŽ. resp. BCK-algebras such that V is a subäriety of U and U is closed under ordinal sums, then the smallest such äriety U is V q. Proof. Any variety U as described in the statement of the corollary must clearly contain V q. It remains to show that V q is closed under ordinal sums. Let A, B g V q. In showing that A q B g V q, we may assume without loss of generality that A and B are finitely generatedŽ since a finitely generated subalgebra of an ordinal sum is an ordinal sum of finitely generated subalgebras of its summands. . By the previous proposi- k m tion, it suffices to show that for any k, m g ␻,ifAgV and B g V n then A q B g V for some n g ␻; the proof is by induction on k. Observe that for any k, the claim is true if all subdirectly irreducible n homomorphic images of A q B lie in some V . But such images either belong to V m or are themselves ordinal sums with lower summand in k k m n V SI, so we need only show that if A g V SI and B g V then some V contains A q B. This is true when k s 0Ž. take n s m q 1 . For k ) 0, by ky1 Proposition 5.3, we have A s C q D for some C g V SI and D g V , n and A q B ( C q Ž.D q B . By the induction hypothesis, D q B g V for n 1 some n g ␻,soAqBgV q . We show that certain important varietal properties are preserved in the passage from V to V q. The next two theorems generalize results obtained for hoops inwx BF . THEOREM 5.7. Let V be a locally finite subäriety of some äriety of pocrims or BCK-algebras that is closed under ordinal sums. Then V q is also locally finite. 306 BLOK AND RAFTERY

Proof. By assumption, V contains, for each r g ␻, only finitely many subdirectly irreducible algebras on at most r q 1 generators, all of which k are finite. Suppose k ) 0 and r g ␻.AnŽ.rq1 -generated algebra E g V k is a subdirect product of algebras C i g V SI, each on at most r q 1 generators. By Proposition 5.3, these C iihave the form A q Bi, where 0 ky1 0 Ai gKsVSI and Biig V .IfB is trivial then C ig K is a finite Ž.subdirectly irreducible algebra and there are at most finitely many

Ž.nonisomorphic C iiof this form. Otherwise, B contains at least one and at most r of the generators of C i and is a subdirect product of algebras ky1 Dij g V SI , each on at most r generators. By the induction hypothesis, these algebras are all finite and there are only finitely many suchŽ noniso- . morphic algebras Dij. Notice that those of the r q 1 generators of C i that lie in Biiiform a generating set S for the algebra B , with <

Bi as described above, hence there are only finitely manyŽ. nonisomorphic Ž. k rq1 -generated C i g V SI. Now the same argument shows that E is k finite. Consequently, by induction, V contains only finitely many Ž.r q 1- generated algebras, all of which are finite, for all k, r g ␻. In particular, r the V q-free algebra on r q 1 free generators Ž.r g ␻ is in V ŽProposi- tion 5.5. , and is therefore finite.

The class of finite algebras in a quasivariety Q shall be denoted by Qfin. Ž. Ž . If Q s ISPPu Qfin , we say that Q is generated as a quasi¨ariety by its PPP finite members. A partial algebra P s ²P; [ , y˙ , 0:²: of type 2, 2, 0 is called a partial pocrim if there is a pocrim A s ²:A; [ , y˙ , 0 with P : A P P P such that [ , y˙˙,0 are the restrictions to P of [, y, 0, respectively, wherever the former are defined. We then call P a partial subpocrim of A. A class K of pocrims has the finite embeddability property Ž.FEP if for every finite partial subpocrim P of any A g K, there exists a finite B g K such that P is a partial subpocrim of B. We implicitly adopt analogous terminol- ogy for BCK-algebras. By a result of Evanswx Eva69 , a quasivariety has the FEP if and only if it is generated by its finite members.

THEOREM 5.8. Let V be a subäriety of some äriety of pocrims or BCK-algebras that is closed under ordinal sums. If V is generated as a quasiäriety by its finite members then so is V q. Ž. q Proof. Assume V s ISPPu V fin . It suffices to show that V has the q FEP, and for this it is enough to prove that V SI has the FEPŽ seew BF, . q Lemma 3.7x . If P is a finite partial subalgebra of A g V SI then A may be m chosen finitely generated and hence in V SI for some m g ␻, by Theorem m 5.5, so we need only proveŽ. by induction on m that each V SI has the VARIETIES OF RESIDUATED POMONOIDS 307

m FEP. By Theorem 5.3, any A g V SI is B q C for some B g V SI and m 1 some C g V y .If ms0, the result follows from the assumption on V . P For m ) 0 and P as above, P l B and Ž P _ Ä0 4. l C are universes of finite partial subalgebras of B and C, respectively, so P l B embeds in a Ž Ä P4. finite BЈ g V SI and, by the induction hypothesis, P _ 0 l C embeds m 1 in a finite CЈ g V y . Then P is embeddable in the finite algebra m BЈ q CЈ g V SI , as required.

6. n-POTENT POCRIMS; THE VARIETIES sq sq MnnAND E

We have seen that for each n g ␻, the varieties Mnnand E are closed under ordinal sums. In this section, we show how the results of Section 5 may be sharpened when we take W to be Mnnor E and V to be the variety generated by all simple algebras in W . A variety V is called a fixedpoint discriminator ¨ariety if for some subclass X of V with V s HSPŽ.X and some ternary term p of V , the following is true: V satisfies pxŽ.,x,yfyand for each A g X, there A exists d g A such that for any a, b, c g A with a / b, paŽ.,b,csd.In this case, p is called a fixedpoint discriminator term for V . Byw BP94, Theorem 3.4x , a variety V generated by simple algebras which has a commutative TD termŽ. see Section 4, Example IV is a fixedpoint discriminator variety for which the TD term is a fixedpoint discriminator term. Every fixedpoint discriminator variety is semisimple, byw BP94, Theorem 3.5x . Recall also that a discriminator ¨ariety is a variety V generated by a class X of algebras, having a ternary term p such that V satisfies A pxŽ.,x,yfyand paŽ.,b,csawhenever A g X, a, b, c g A, and a / b. In this case V is semisimple and arithmeticalwx BS81, Theorem IV.9.4 . Let n g ␻ be given. The following result may be considered an extension of Theorem 3.4.

THEOREM 6.1.Ž. i If K is any class of simple Ž n q 1- .potent pocrims then HSPŽ.K is a discriminator äriety of pocrims. Ž.ii If K is any class of simple BCK-algebras in En then HSPŽ.K is a semisimple äriety of BCK-algebras. In particular, eëry finite set of finite simple pocrimsŽ. resp. BCK-algebras generates a discriminatorŽ. resp. semisimple äriety of pocrims Ž resp. BCK- algebras..

Proof. Ž.i Define tx Ž,y,z .swzy˙˙˙˙nxŽ.Ž.yyynyyxxw[ xy ˙˙Žxy nxŽ.Ž..y˙˙yynyy ˙x x. Let A g K. Since A is a pocrim, A satisfies txŽ,x, z.fz[ Žxy˙ x .fz Žusing Ž. 2 , Ž. 6 , and z [ 0 f z.. Since A is simple, it 308 BLOK AND RAFTERY has no nontrivial ideals, so if a, b, c g A with a / b then ²:a y˙˙b, b y a A sA, so that c y˙˙˙˙kaŽ.Ž.ybylbyas0 for some k, l g ␻ Žsee the characterization in Section 2 of ideals generated by subsets. . But this implies that c y˙˙˙˙naŽ.Ž.yb ynbyas0, since A is Ž.n q 1 -potent. Thus, Asatisfies x fu y ª txŽ.,y,zf0[ Ž.xy˙ 0fx. This means that t is a Ž. Ž. discriminator term for HSP K and of course, HSP K : Mn : M. Ž.ii Recall that pxŽ.Ž.Ž.,y,zszy˙˙˙˙nxyyynyyx is a commutative TD term for En, and therefore for HSPŽ.K , so Ž. ii is a special case of the remarks preceding this theorem.Ž In this case, we may take d s 0 through- out K.. The last statement follows because every finite pocrimŽ resp. BCK- .Ž. Ž. algebra satisfies Ennfor some n g ␻ and the varieties M resp. En, ng␻, form an ascending chain.

Inasmuch as the argument ofŽ. ii also proves the semisimplicity of HSPŽ.K in the case where K is a class of simple Žn q 1 . -potent pocrims, Theorem 6.1Ž. i illustrateswx BP94, Theorem 3.8 , which says that the discriminator varieties are just the congruence permutable semisimple varieties with EDPC. Of course, we cannot extend Theorem 6.1Ž. i to the case where

K is a class of simple algebras in En, since no nontrivial variety of BCK-algebras is congruence permutable. Theorem 6.1Ž. i may be used to show that not e¨ery congruence permutable ¨ariety of pocrims consists of hoops. For example, consider theŽ. unique pocrim A s ²A; [ , y˙ , 0 : , with AsÄ40, 1, a, 2 satisfying 0 - 1 - a - 2 and x [ y s 2 for all x, y G 1. Since A is finite and simple, HSPŽ. A is a discriminator variety Ž and therefore congruence permutable. but A itself is not a hoop, since 1 - a but there is no b g A such that 1 [ b s a. Theorem 6.1 and its proof also suggest that simple algebras in Mnnor E are ‘‘nicely structured.’’ Of course, a minimalŽ. nonzero ideal of an algebra in Mnnor in E is just such a simple algebraŽ. by the congruence extension property . The following result will be useful:

LEMMA 6.2. Let A be a subdirectly irreducible algebra in Mnnor in E and let I be itsŽ. unique minimal nonzero ideal. Then²: A; F has a unique atom Ä4 a,and I s c g A: c y˙ na s 0. In the case that A g Mn, I is the principal order idealŽ bx s Ä4c g A: c F bofA²:;F,where b s na, and for any cgA,we ha¨ecfI iff nŽ. c y˙ b G b.

Proof. Pałasinskiwx Pał a, Theorem 2 has shown that if B is any subdi- ²: rectly irreducible BCK-algebra in the variety En then the poset B; F has a unique atom. In particular, therefore, ²:A; F has a unique atom a, and a g I. Since I is the universe of a simple subalgebra of A, the ideal I is generated by a, and since A is Ž.n q 1 -potent, I must have a greatest VARIETIES OF RESIDUATED POMONOIDS 309 element, viz., b s na.If cgARIthen c y˙˙b / 0, hence c y bGaso that ncŽ.y˙˙b Gna s b.If cgIthen c y b s 0soncŽ.y˙b s 0Gub. s s We define Mnnand E to be the varieties generated, respectively, by the class of all simple Ž.n q 1 -potent pocrims and the class of all simple algebras in En. For the remainder of this section, V and W shall denote ssŽ. either Mnnnnand M , or E and E for a fixed but arbitrary n g ␻ .By 0 Theorem 6.1, V is semisimple, so V SI s K is the class of all simple m Ä 0 my14 m algebras in V , and K s A q B: A g K and B g V s V SI for all mg␻Ž.Corollary 5.4 . Also, V qis the smallest variety U such that for every simple A g V , we have A q B g U whenever B g U Ž.Corollary 5.6 . These results and Proposition 5.5 offer model theoretic descriptions of the variety V q. We shall now obtain equational characterizations of V q Žin . both of the cases V : Mnnand V : E . Consider the following identitiesŽŽ recall that an inequality tx,y.F sxŽ.,y abbreviates the identity txŽ.Ž.,yy˙sx,yf0: .

␴Ž.Ž.x,y : x[ny y˙˙˙x F Ž.Ž.ny y nxyny ;

␤Ž.x,y:xy˙˙˙ Žxyy .ynxwy ˙˙˙yyŽ.xyny xF y y ˙nxŽ.y ˙y;

␥Ž.x,y:yy˙˙xynxy ˙˙yyŽ.Ž.xy ˙ny F y y ˙nxy ˙y. By way of motivation, note that ␴ Ž.x, y implies Ž over M .the quasi-identity

␴ ЈŽ.x, y : ny F nx Žy˙ny .ª x [ ny F x, which is equivalentŽ. over Mn to the quasi-identity

␴ Љ Ž.x, y : y [ y f y & y F nxŽ.y˙y ª x[yfx, Ž. Ž. since Mn satisfies y [ y f y m y f ny. Also, ␤ x, y & ␥ x, y implies Ž.over BCK

␣ Ž.x, y : x y˙˙y f x y 2 y & y y ˙˙nxŽ.yyf0

ªxy˙˙yfx&yyxf0

Ž.which is equivalent to the conjunction of two quasi-identities . We shall see presently that ␴ Ž.x, y and ␴ Љ Ž.x, y are equivalent over Mn, and that ␤Ž.x,y&␥ Ž.x,yis equivalent to ␣Ž.x, y over En. Note that every pocrim satisfies both ␴ Ž.x, 0 and x F y ª ␴ Žx, y .Žby definition of residuation.Ž. . Also every BCK-algebra satisfies both ␤ x,0 & ␥Ž.x, 0 and x F y ª w␤Ž.x, y & ␥ Ž.x, y x. 310 BLOK AND RAFTERY

PROPOSITION 6.3. V q satisfies ␤Ž.x, y and ␥ Ž.Žx, y hence ␣ Ž..x, y , sq and Mn satisfies ␴ Ž.Žx, y hence ␴ Љ Ž..x, y . Proof. For the first assertion, it suffices to show that every subdirectly irreducible 2-generated member of V q satisfies ␤Ž.x, y & ␥ Ž.x, y .By 1 Proposition 5.5, we need only show that V SI satisfies ␤Ž.x, y & ␥ Ž.x, y . Moreover, by the remarks preceding this proposition, we need only show 1 Ž. Ž. that V SI satisfies wxwy fu 0&xFuy ª ␤ x,y &␥ x,yx. Similarly, for Ž s.1 the second assertion, it suffices to show that Mn SI satisfies w y fu 0& xFuyx ª␴Ž.x,y. 1 Let C g V SI and a, b g C with b / 0 and a Fu b. First suppose that 0Ž. Ž. CgVSI sKV. By simplicity of C, a y˙˙˙nb s 0 and c y nayb s0 for all c g C. Thus, a y˙˙Ž.a y b y ˙nay ˙˙˙by Ž.aynb s 0 s b y ˙˙a y nay ˙˙˙byŽ.aynb so ␤Ž.a, b and ␥ Ž.a, b are true. If in addition, C is a pocrimŽ e.g., if s. V s Mn then

Ž.a [ nb y˙˙˙˙a F nb s nb y 0 s Ž.Ž.nb y naynb , so ␴ Ž.a, b is true. This shows that V satisfies ␤Ž.x, y & ␥ Ž.x, y and that s Mn satisfies ␴ Ž.x, y . Now suppose that C f V SI . By Corollary 5.4, C s A q B, where A g 0 KŽ.V,BgV, and both A and B are nontrivial. In view of the foregoing arguments, we may assume that neither A nor B contains Ä4a, b . It follows that b g A and a g B, hence b F a Ž.i.e., b y˙˙a s 0 and a y b s a Žso ay˙˙Ž..aybs0 . The left-hand sides of ␤Ž.Ž.a, b and ␥ a, b are therefore both 0, so ␤Ž.a, b and ␥ Ž.a, b are true. If C is a pocrim then nb g A Žsince Ais an ideal of C.Ž.,so a[nb s a, hence a [ nb y˙ a s 0. This is the left-hand side of ␴ Ž.a, b ,so␴ Ž.a,b is true.

PROPOSITION 6.4. Let A be a subdirectly irreducible pocrim or BCK- y Ž. algebra with A g En. If A satisfies ␣ x, yorAis a pocrim satisfying ␴ЉŽ.x,y then A is isomorphic to an ordinal sum I q D, where I is the least nonzero ideal of A. Proof. We may assume that n ) 0. Suppose first that A satisfies ␣Ž.x,y. Let I be the least nonzero ideal of A and let D s Ž.A R I j Ä40. Let b g I and a, c g D. We know that I is a subuniverse of A. We want to show that b F a, that a y˙˙b s a, and that a y c g D unless a F c. Using Ž.En , we have

a y˙˙˙˙Ž.n y 1 b y b s a y nb s a yŽ.n q 1 b s a y ˙˙˙ Ž.n y 1 b y b y b. VARIETIES OF RESIDUATED POMONOIDS 311

Also, a y˙ kb f I Ž.because a f I for all k G 0, so the ideal of A generated by a y˙˙kb contains I, whence b y naŽ.Žy˙kb s 0 see Lemma 6.2 . . So

by˙˙nayŽ.ny1by ˙b sby ˙˙nawxynb s 0.

By ␣ŽŽa y˙ n y 1 ..b, b ,

a y˙˙˙Ž.n y 1 b y b s a y Ž.n y 1 b Ž.i.e., a y ˙˙nb s a yŽ.n y 1 b and b y˙˙wa yŽ.n y 1 bxs 0, i.e., b F a y˙Ž.n y 1 b.If ns1, these are the required results about a and b. If n G 2, we repeat this argument, replacing a y˙˙Ž.n y 1 b by a y Žn y 2.Žb, and obtain a y˙˙nb s a y n y 2.b and, eventually, a y˙˙nb s a y b. In particular, a y˙˙˙˙˙b s a y b y b and b y naŽ.yb s0, so by ␣ Ž.a, b , ay˙˙bsaand b y a s 0, i.e., b F a. Now if a Fu c, we claim that a y˙c g ARI. For if a y˙˙c g I then a s a yŽ.Ža y˙c by the above . , implying aFc! This shows that A is the ordinal sum of I and D. Ž. Ž. Now suppose A is a pocrim hence A g Mn satisfying ␴ Љ x, y .By Lemma 6.2, I s Žbx, with b s na, where a is the unique atom of ²:A; F . Let wb. s Ä4c g A: b F c . Notice that if c g A and c Fu b then ncŽ.y˙b G na s b, whence c [ b s c,by␴ЉŽ.x,y, from which b F c follows. Thus, AsŽbxwjb., and A is the ordinal sum of the subalgebras with universes Žbxwand b.j Ä40.

THEOREM 6.5.Ž. i The identities Ž.Ž.Ž.Ž1, 2, 3, En .,␤ Žx,y .,and ␥ Žx, y . sq form an equational base for En . Ž.ii The identities Ž.1, Ž. 2, Ž. 3, Ž. 4, ŽMn ., and ␴ Žx, y . constitute an sq equational base Mn . Proof. The following argument proves bothŽ. i and Ž ii . : let V and W be s s Ennand E inŽ. i , and M nand M ninŽ. ii . Let X be the variety axiomatized by the identities in the statement of the theorem. Then V q: X, by Proposition 6.3 and the fact that V q: W . Conversely, let A be a finitely generated member of X. In showing that A g V q, we may clearly assume without loss of generality that A is subdirectly irreducible. By Proposition 6.4, we may assume that A is isomorphic to an ordinal sum I q D, where I is the least nonzero ideal of A, and D g X. At least one generator of A lies in I, by the definition of 0 ordinal sums. If A is 1-generated then D is trivial and A g K Ž.V : V q. Suppose A is r-generated, where r ) 1. Then D ( ArI is Ž.r y 1 -generated Ä4 and is a subdirect product of a family Bj: j g J of subdirectly irreducible members of X, each of which is Ž.r y 1 -generated. Applying Propo- q sition 6.4 and an induction hypothesis to the Bj, we deduce that D g V , hence A g V q, by Corollary 5.6. We have shown that V q contains all 312 BLOK AND RAFTERY finitely generated members of X. Since every algebra is embeddable into an ultraproduct of its finitely generated subalgebras, it follows that X : V q. The next result is an immediate corollary but would appear to be difficult to derive directly. Ž. Ž. Ž. Ž. COROLLARY 6.6. i En * w␤ x, y & ␥ x, y xl ␣ x, y . Ž. Ž. Ž. ii Mn * ␴ x, y l ␴ Љ x, y . Ž. Ž. Ž. Ž. iii Mn * w␤ x, y & ␥ x, y xl ␴ x, y . The preceding construction and results extend earlier work on Brouwer- sq ian semilattices and hoops. It is known and easy to see that M11s M and sqqŽ E11sE. The variety H of hoops coincides with J0recall that J0is the variety of all commutative pocrims, i.e, the variety generated by all linearly ordered commutative pocrims. : this iswx BF93, Theorem 3.3 combined with our Proposition 4.1 and it yields a proof of the fact that H is generated by its finite memberswx BF93, Corollary 3.6 .Ž The result illustrates Theorem 5.8..Ž Recall that hoops and, in particular, Brouwerian semilattices . are congruence permutable. Similarly, we have: sq Ž PROPOSITION 6.7. The ¨ariety Mn is congruence permutable hence arithmetical..

sq Proof. In view of Theorem 3.4, it suffices to show that Mn satisfies the identity

␦ Ž.x, y : y y˙˙˙˙nx Žyy .yny Žyx . [xy˙˙Ž.xynxŽ.Ž.y ˙˙yynyy ˙x fx, and for this purpose it suffices to consider subdirectly irreducible 2-gener- s ated algebras. It is easy to see that every pocrim satisfies ␦Ž.x, x . Since Mn is generated by simple Ž.n q 1 -potent pocrims, it satisfies

x fu y ª z y˙˙˙˙nxŽ.Ž.yyynyyxf0, and therefore also ␦Ž.x, y . Now suppose that C s A q B, where A g 0Ž s. s K Mnnand B g M , and let a, b g C with a / b. If both a and b are in Aor both are in B then ␦Ž.a, b follows from the case just considered. If a g A and b g B then a F b and b y˙ a s b, so the left-hand side of ␦Ž.a,bevaluates to wxwb y˙˙˙˙nb [ a yŽ.a y nb xs 0 [ Ž.a y 0 s a,asre- quired. If a g B and b g A then b F a and a y˙ b s a so the left-hand side of ␦Ž.a, b becomes wxwb y˙˙˙˙na [ a yŽ.a y na xs 0 [ Ž.a y 0 s a. This completes the proof, in view of Propositions 5.3 and 5.5. VARIETIES OF RESIDUATED POMONOIDS 313

On the other hand, we have:

PROPOSITION 6.8. For each n G 2, the ¨ariety Mn is not congruence sq permutable, hence Mnn/ M .

Proof. It suffices to show that M2 is not congruence permutable. Let ²:A; F be the poset depicted in Figure 1; A s I j S, where Ä4 Ä 4 Is0, 1, 2 and S s a00, a 10, a 11, a 21 . Define a binary [ on A by

x [ y s y [ xxŽ.,ygA, i[jsminÄ4i q j,2 Ž.i, jgI ,

aij [ksaaiy˙˙k, jykijŽ.gS,kgI,

aij[a kl saa00 Ž.ij,a kl gS.

Then ²:A; [ ,0;F is a commutative integral pomonoid, which is residuated as follows:

x F y « x y˙˙y s 0, a00y a 21 s 2, x y˙˙0 s x, x y y s 1 for all other pairs x, y g S with x ) y,

2 y˙˙1 s 1, aij y k s aminÄiqk,24,minÄjqk,14 Ž. aij gS,kgI.

The resultant pocrim A s ²:A; [ , y˙, 0 clearly satisfies 2 x f 3 x,soAg Ž . ŽŽ . ÄŽ.Ž.4. M21. The relation ␶ s I = I j S = S R a 1, a10, a10, a11is a toler-

FIGURE 1 314 BLOK AND RAFTERY ance on A which is not transitive, i.e., ␶ f Con A. Thus, tnŽ.A ) 1. In view of the remarks following Lemma 1.1 and the fact that M2 is congruence Ž. Ž . 3-permutable, we conclude that tn A s 2 s tn M22,so M is not congruence permutable.Ž. Ironically, A is congruence permutable.

sq We remark that M2 is not the largest congruence permutable subvariety of M2 . Let A be the unique pocrim whose underlying poset is the chain Ä40, 1, a, 2 with 0 - 1 - a - 2 such that 1 [ 1 s 1 and x [ y s 2 for all x ) 1 and y G 1. It is easily checked that

txŽ.,y,zszy˙˙˙˙2 Ž.Ž.xyyy2yyx [xy˙˙Ž.xy2Ž.Ž.xy ˙˙yy2yy ˙x is a Mal’cev term for HSPŽ. A , i.e., A satisfies txŽ.Ž.,x,yfyfty,x,x,so Ž. Ž. sq HSP A is a congruence permutable subvariety of M22. But HSP A M ; sq Ž. in fact, A f M2 . To see this, it suffices to show that A ^ ␴ Љ x, y . Indeed, 1 [ 1 s 1 and 2Ž.a y˙ 1 s 2 a s 2 ) 1, but a [ 1 s 2 / a. The fact that M2 is not congruence permutable brings clarity to a question concerning varieties with EDPC. Consider a variety K with a constant term 0. We say that K is Fregean with respect to 0Ž briefly, 0-Fregean. if K is 0-regular and

A A ⌰ Ž.a,0 s⌰ Ž.b,0 « asb holds for all A g K and all a, b g A. A binary term txŽ.,yis called a weak meet with respect to 0Ž. briefly a weak 0-meet if

A A A ⌰ Ž.t Ž.a, b ,0 s⌰ Ž. Ž.Ž.a,0 , b,0 for all A g K and all a, b g A. These definitions are taken fromwx BKP84 , whose Corollary 3.10 shows that a 0-Fregean variety K with EDPC has a weak 0-meet iff K is congruence permutable. In the remarks that follow this corollary, it is claimed that the implication from left to right would fail were the requirement dropped that K be 0-Fregean. P. Idziak pointed out that the justification given inwx BKP84 for this claim is flawed. The claim is vindicated, however, by a consideration of the variety M2 . Indeed, we have noted that each of the varieties Mn has EDPC; as pocrim varieties, they Ž. are 0-regular and tx,ysx[yis clearly a weak 0-meet, but M2 is not congruence permutable. Ž y. We have made repeated use of the fact that S Mnn: E . It is an open problem whether every BCK-algebra in En is a residuation subreduct of an Ž. Žy. nq1 -potent pocrim, i.e,. whether S Mnns E . A partial result in this direction is the following. VARIETIES OF RESIDUATED POMONOIDS 315

sq THEOREM 6.9. For each n g ␻, the ¨ariety En is just the class of sq ŽŽ sq.y. sq residuation subreducts of the ¨ariety Mnnn, i.e., S M s E . Žy. Proof. Since S Mnn: E and nontrivial subalgebras of simple BCK- ŽŽ 0Ž s..y. 0Ž sq. s algebras are simple, we have S K Mnnn: K E . Since M and s ŽŽ Ž ..y.ŽŽy.. En are semisimple and SPssL sPSL for any class L of ŽŽ s.y. s pocrims, it follows that S Mnn: E . 0Žs. Let A g K Enn, i.e., A is a simple algebra in E . We show that Žy.²: AgSB for some simple algebra B g Mn. By Lemma 6.2, A; F has a unique atom, a. By simplicity, we have b y˙ na s 0 for all b g A. Since a is an atom, this means that b y˙˙˙a1y иии yans 0 for all b g A and all nonzero a1,...,an gA. We shall use the notational conventions of Theo- rem 2.1. Observe that ²Ž.J A ; : :also has a unique atom, viz., I s JaŽ.s Ä40, a , and nI s A, where nI is I q иии qInŽ.summands . This means that for any J g JŽ.A ,if J/Ä40 then nJ s A. It follows that for any proper order filter of ²Ž.J A ; : :, say F g F ŽŽ..JA , we have nF s J Ž.A , whence nF s Žn q 1 .F. Of course nJ Ž.A s J Ž.A s Žn q 1 .Ž.J A also, so the pocrim CsFJAŽŽ ..is Žn q 1 . -potent. We also have that G s ÄJ g JŽ.A : Ja Ž.:J4 is the unique atom of ²:C; = ,soCis a simple algebra. By Theorem 2.1, A y s is isomorphic to a subalgebra of C , as required. Since En is semisimple ŽŽŽ 0Ž s..y.. ŽŽ Ž 0Ž s...y. s ŽŽ s.y. and PSs K Mn :SPs K Mnnn, we obtain E : S M , sŽŽ s.y. and so Enns S M . Žs.mŽŽŽ s.m.y. We claim that Enns S M for any m g ␻. Since the residuation subreducts of the ordinal sum of a pair A, B of pocrims are just the ordinal sums of pairs of residuation subreducts of A, B, it follows from Ž s.m ŽŽŽ s.m .y. Corollary 5.4 and the induction hypothesis that En SI s S Mn SI . The subdirect product argument used in the case m s 0 therefore shows s m s m y that ŽEnn. and SŽŽŽ M . . . have the same subdirectly irreducible mem- sq bers, completing the proof of our claim. By Proposition 5.5, En and sq y SŽŽ Mn . . have the same finitely generated algebras, so we must have sq ŽŽsqy . . Enns S M .

sq sq 7. THE VARIETIES M22AND E ; LOCAL FINITENESS

Žsq.Žsq. We have mentioned that M11s M and E 11s E are locally finite varieties and that the varieties Mnnand E are not locally finite for any sq sq nG2. In this section, we show that M22and E are also locally finite s s and that M22and E have continuously many subvarieties. s s s LEMMA 7.1. The ¨arieties M22and E are locally finite. The ¨ariety M3is not locally finite.

Proof. Let A be a finitely generated simple pocrim in M2 , let 1 be the 316 BLOK AND RAFTERY

Ž.unique atom of ²:A; F and let 2 s 1 [ 1. Suppose that the generators of A other than 0, 1, 2 are x1,..., xri. By simplicity, we must have 1 - x - 2 for all i, and by compatibility of the partial order with the monoid operation, also xij[ x s 1 [ x is 2 for all i, j. Thus, xijy˙ x s 1 whenever xijFu x . This means that <

x [ y s y [ x, x [ 0 s x,

1 [ 1 s bi [ 1 s 2,

bij[ b s a?Žiqj.r2@, xG1, y G 2 « x [ y s 3.

It is not hard to see that ²:A; [ ,0;F is a commutative integral pomonoid. It is also residuated as follows:

x F y « x y˙ y s 0, x y˙ 0 s x,

biiy˙˙˙1 s 2 y 1 s 2 y b s aiiy ˙˙˙2 s 3 y 2 s 3 y a s 1,

aiiy˙˙˙1s3y1s3ybs2, 0, if i G j, a ˙˙a b b ijijy s y s ½1, if i - j,

1 2, if 2 j ) i, aijy˙ b s 1 ½bk, where k s maxÄ4kЈ g ␻ : 2Ž.j q kЈ F i , otherwise. VARIETIES OF RESIDUATED POMONOIDS 317

FIGURE 2

Thus, A s ²:A; [ , y˙ , 0 is a pocrim. Clearly, A satisfies 3 x f 4 x, since 3c s 3 whenever 0 / c g A. Also, A is simple, since 1 [ 1 [ 1 s 3. We 0Ž s. Ä4 therefore have A g K M30and A is generated by b , since biii[ b s a , aiiiy˙˙bsbq1,biiybq1s1, 1 [ 1 s 2, and 2 [ 1 s 3. sq sq THEOREM 7.2. The ¨arieties M22and E are locally finite. Proof. This follows from the previous result and Theorem 5.7.

sq By Lemma 7.1, Mn is not locally finite for any n ) 2. We do not know sq whether any of the varieties En Ž.n ) 2 are locally finite. It clearly s suffices to settle this question for the varieties En . 318 BLOK AND RAFTERY

s s / 0 THEOREM 7.3. Each of the ¨arieties M22and E has 2 distinct sub¨arieties. Proof. By the proof of Lemma 7.1, the subdirectly irreducibleŽ i.e., s simple. algebras in M2 are, up to isomorphism, just the pocrims A P , where Ä4 ²: APPs0, 1, 2 j P, P; F is any poset, the pocrim order F contains FP ,0-1-p-2 for all p g P, and a [ b s 2 whenever 1 F a, b g A. ŽŽ.. Ž. We allow P s л. See Fig. 3 . It is easy to see that A PQg IS A iff ²:Ž. ² : P;FPQis order- embeddable in Q; F . Now consider the sequence of posets K12, K , . . . , where K nis the 2Ž.nq1 -element ‘‘crown’’ depicted in Figure 3. It is easy to see that for n, m 1, K is a subposet of K iff n m. Writing A for A , we have G nms nKn Ž. Ž. AnmfIS A whenever n / mn,mG1 . For each n G 1, the assertion ‘‘there is no subalgebra isomorphic to A n’’ is equivalent to a first-order ²: Ž sentence in the language [ , y˙ , 0 , which is valid in every A m n / m G . Ä4 Ž. 1 . Thus if B : A mn: m G 1 and A f B, we have A nf ISPuB s HSPuŽ.ŽB by semisimplicity and the fact that nontrivial subalgebras and ultraproducts of simple Ž.n q 1 -potent pocrims are simple. . By Jonsson’s´ Ž. Ä theorem, A n f HSP B . Thus, if B and C are distinct subsets of A m: 4 Ž. Ž. Ä4/0 mG1 then HSP B / HSP C . The fact that A m: m G 1 has 2 s subsets completes the proof, as far as M2 is concerned. s s For E2 , the argument requires only minor modification, since E2 is also semisimple and congruence distributive, nontrivial subalgebras of simple s BCK-algebras are simple, and the subdirectly irreducible members of E2 s are just the residuation reducts of those of M2 , together with the subalgebras of these that are obtained by removing the top element.Ž. See Fig. 3.

/ 0 This means that M22and E each have 2 distinct semisimple subvarieties, contrasting with the fact that M11and E each have just one semisimple subvarietyŽŽ viz., the varieties G and S Gy...

FIGURE 3 VARIETIES OF RESIDUATED POMONOIDS 319

Ž. In the previous proof, the fact that A nnf HSP B whenever A f B : Ä4 Ai:is1, 2, . . . is actually an instance of a more general phenomenon. Bywx BP82, Corollary 3.2 , in a variety V of finite type which has EDPC, every finite subdirectly irreducible algebra A is splittingŽ i.e., there is a largest subvariety of V not containing HSPŽ.. A . Since M22and E have Ž. Ž.Ž Ž..Ž . EDPC and A nmf IS A s HS A m= HSP A mSI Jonsson’s´ theorem , Ž.Ž . Ž. i.e., A nmf HSP A n / m , we may infer A nf HSP B from the fact that A n is splitting.

8. VARIETIES OF CANCELLATIVE POCRIMS

It is natural to consider pocrims A whose underlying monoids are cancellative. In this section we investigate varieties of such ‘‘cancellative’’ pocrims. First recall the identity

Ž.x [ y y˙y f x Ž.18 and consider the quasi-identities

x [ y F x [ z « y F z Ž.20 x[yfx[z«yfz.21Ž.

LEMMA 8.1. Conditions Ž.Ž.18 , 20 , and Ž.21 are equi¨alent in any pocrim. Proof. Since F is a partial order with respect to which a map a¬ay˙ xis isotone for any x, we clearly have thatŽ. 18 « Ž.20 « Ž.21 . It is an easy consequence ofŽ. 7 that Ž 21 .« Ž18 . . We shall say that a pocrim A is cancellati¨e if it satisfies the equivalent conditionsŽ.Ž. 18 , 20 , and Ž. 21 , i.e., if the underlying monoid ²A; [ ,0 : is cancellative. We denote by C the class of all cancellative pocrims, which is clearly a quasivariety. Since a finite pocrim must have a greatest element, it is easy to deduce fromŽ. 18 that every nontrivial cancellative pocrim has an unbounded underlying poset and is therefore infinite. More generally,

C l Mn is the trivial variety, for each n g ␻. Observe that if A g C, then the term txŽ.Ž.,y,zs x[zy˙ yis a Mal’cev term for A, i.e., A satisfies txŽ.,x,yfyand tx Ž.,y,yfx. We therefore have: PROPOSITION 8.2. If A is a cancellatië pocrim then the tolerance number of A is 1, hence A is congruence permutable. Eëry äriety of cancellatië pocrims is arithmetical and 0-regular, with the congruence extension property. A Proof. Let ␶ g Tol A and Ž.Ž.a, b , b, c g ␶.Then ŽtaŽ.,b,b, A taŽ..Ž.,a,cg␶, i.e, a, c g ␶. Thus, tn Ž.A s 1, so A is congruence permutable. Now Proposition 3.1 remains true if we set K s C. This estab- lishes the remainder of the result. 320 BLOK AND RAFTERY

Inwx BR93 , we constructed a pocrim A and a subalgebra B of A such that ²:A;[, 0 is a cancellative monoid and there is a congruence ␩ on B such thatŽ. i Br␩ is not a pocrim, and Ž ii . there is no congruence ␪ on A such that ␩ s ␪ l Ž.B = B . Thus, in contrast to the previous result, we have: PROPOSITION 8.3. The quasiäriety C of all cancellatië pocrims is not a äriety and does not enjoy the congruence extension property. Varieties of cancellative pocrims are characterized by the next result, which is an immediate corollary of Theorem 3.2 and Proposition 8.2:

PROPOSITION 8.4. Let H be a class of algebras of type ²:2, 2, 0 . Then HSPŽ.K is a ¨ariety of cancellati¨e pocrims iff there exists a 6-ary ²[ , y˙ :- term such that K satisfies the identities Ž.Ž.Ž.Ž.Ž1, 2, 3, 4, 18, .xftx Ž ,y,xy˙ y,yy˙˙x,0,0,.Žand t x, y,0,0, xyy, yy˙x.fy. For an arbitrary variety V of pocrims, it is not clear whether we necessarily have HŽV y. : BCK. The situation for varieties of cancellative pocrims is clearer:

PROPOSITION 8.5. Let A be a cancellatië pocrim. Then Con A s Con Ay. Consequently, if HAŽ.:M, e.g., if A generates a äriety of Ž cancellatië . pocrims, then HAŽ y. :BCK. Proof. Let ␪ g Con Ay and let a, b, c, d g A such that a ␪ b and c ␪ d. Then byŽ 18 . , a [ c s ŽŽa [ c [ b .y˙˙b .␪ ŽŽa [ c [ b .y a . s c [ b. Similarly Ž.Ž.c [ b ␪ b [ d and the result follows from the transitivity of ␪. The poset P¨ Ž.M of all varieties of pocrimsŽ ordered by inclusion. is a nearlattice, i.e., a lower semilattice in which any finite subset which is bounded above has a supremum. In addition, it is clearly closed under arbitrary nonempty intersections, but it has no greatest elementw BR93, Theorem 3x . Also, since pocrim varieties are congruence distributive, it follows easily that P¨ Ž.M is a distributië nearlattice, i.e., one in which the meet operation distributes over all existent finite suprema. It is not clear, however, whether P¨ Ž.M is a lattice.Ž It clearly suffices to establish whether the union of any pair of pocrim varieties is contained in a pocrim variety.. The analogous question for varieties of BCK-algebras was an- swered affirmatively inwx BR95, Theorem 11 , but the methods used there do not generalize readily to the case of pocrim varieties. Nevertheless, for different reasons, the corresponding result about cancellative pocrims is true:

THEOREM 8.6. The poset P¨ Ž.C of all ärieties of cancellatië pocrims is q aŽ. distributië lattice. In fact, it is a sublattice of the lattice L Ž.C of all subquasiärieties of C. VARIETIES OF RESIDUATED POMONOIDS 321

Proof. We have observed that txŽ.Ž.,y,zs x[zy˙ yis a Mal’cev term for all cancellative pocrims, and therefore also for the variety HŽ.C generated by C. It follows that HŽ.C is congruence permutable, hence congruence modular. Let K, L g P¨ Ž.C . Hagemann and Hermannw HH79, Corollary 4.3x proved that in the subvariety lattice of a congruence modular variety, the join of two congruence distributive subvarieties is congruence distributive. In LH¨ Ž ŽC .., therefore, the join N s K k L s HSPŽ K j L . is a congruence distributive variety. By a well-known consequence of Ž. Jonsson’s´ theorem, NSIs K SIj L SI. Since N s IPs NSI , we have N : Ž. ISPPu K j L : C. It follows immediately that N is the join of K and L ¨ Ž. Ž. qŽ. in P C . Of course, the join ISPPu K j L of K and L in L C must be Ž. ¨Ž. contained in N,so NsISPPu K j L . Since P C is also closed under intersections, this shows that P¨ Ž.C is a sublattice both of LH¨ŽŽC ..and of LqŽ.C. Henceforth, we shall write L¨Ž.C in place of P¨Ž.C . The most obvious example of a cancellative pocrim is the algebra of natural numbers ␻. In fact the variety ISPPuŽ.␻ is just the class of all cancellative hoopswx BF93, Corollary 2.4 . As an immediate consequence of Theorem 4.1, we therefore have:

PROPOSITION 8.7. The äriety ISPPuŽ.␻ is just the class of all commuta- tië cancellatië pocrims, i.e., it is axiomatized by Ž.Ž.Ž.Ž.Ž1, 2, 3, 4, 18, . and Ž.T.

We have already mentioned that ISPPuŽ.␻ is an atom of the poset P¨ Ž.Mof varieties of pocrims, so it is also an atom of L¨Ž.C . Its position in L¨ Ž.Cmay be described more precisely with the aid of a technical lemma involving the following weaker variant of the cancellative lawŽ. 18 : Ž.x [ x y˙x f x.22Ž.

LEMMA 8.8. A nontri¨ial quasi¨ariety V of pocrims satisfies the identity Ž.x[xy˙ xfx iff the V-free algebra on one free generator is isomorphic to ␻. Proof. Sufficiency is obvious since ␻ satisfiesŽ. 22 . Conversely, suppose that V satisfiesŽ. 22 . Let n, m g ␻ with m - n. We know that M, and hence V , satisfies the identities mx F nx and nx [ mx f Ž.n q mx. Also, Vcannot satisfy mx f nx. Indeed, if V satisfies mx f Ž.m q 1 x then m)0 and V must satisfy mx f kx for all k G m, whence, byŽ. 22 , V also satisfies mx f Ž.Ž.2mx y˙ mx f 0, contradicting nontriviality. It therefore suffices to check that V satisfies Ž.Žnx y˙ mx .f Žn y mx .. We may assume m ) 0. From elementary properties of pocrims it follows that M satisfies Ž.Žnx y˙˙mx .F Ž.Ž. Žn y lx .y Žmylx . Ž.23 322 BLOK AND RAFTERY whenever m G l g ␻. In particular, M satisfies Ž.Žnx y˙mx .F Žn y mx .. For the reverse inequality, consider two cases. If n G 2m then for l s n y 2m, V satisfies

Ž.n y mxfŽ.Ž.Ž.2 Ž.nymxy˙ Ž.nymx FŽ.Ž.Ž.Ž.2Ž.nymylxy˙ Žnymylx .by Ž. 23 fŽ.Žnx y˙mx ..

k If n - 2m, choose k to be the first positive integer such that 2 Ž.n y m G k k n. Set l s 2 Ž.n y m y n F Ž2 y 1.Žn y m .. By repeated application of Ž.22 , V satisfies

k k1 Ž.n y mxfŽ.Ž.Ž.2 Ž.nymxy˙Ž.2y Ž.nymx y˙˙иии yŽ.Ž.Ž.2Ž.n y mxy˙ Ž.nymx

ky1 ki fŽ.Ž.2Ž.nymxy˙Ý2 Ž.nymx/ ž/ž/i0 s k k fŽ.Ž.2Ž.nymxy˙Ž. Ž.Ž.2y1nymx

k k FŽ.Ž.2Ž.nymylxy˙Ž.Ž. Ž.Ž.2y1 nymylx Ž.by Ž. 23 fŽ.Žnx y˙mx ., which completes the proof.

COROLLARY 8.9. The äriety ISPPuŽ.␻ is the smallest nontriïal quasiä- riety of cancellatië pocrims. Proof. SinceŽ. 18 implies Ž. 22 , the previous lemma shows that the algebra ␻ must belong to every nontrivial quasivariety of cancellative pocrims. We shall show that, in contrast with Corollary 8.9, there is no largest variety of cancellative pocrims. In the process we use the varieties Jn l C, n g ␻, where Jn is as defined in Example VI of Section 4. Ä4Ä4 For each k g ␻, let Aks 2n: n s 0,...,k, ng␻ j ng␻:2k-n. ²: Then the commutative integral linearly ordered monoid Ak; q,0;F Ž.where q is natural addition and F is the natural linear order is a submonoid of ␻ satisfying the conditions of Lemma 2.2, and is therefore VARIETIES OF RESIDUATED POMONOIDS 323 residuated. The residuation operation y˙k may be defined as follows: we have n y˙ k m s 0if nFm, while for n G m:

n y m,ifnymis even or n y m G 2k, n y˙k m s ½ n y m q 1, otherwise. ²: Ž. By Lemma 2.2, A kkks A ; q, y˙ , 0 is a pocrim. Since A ksatisfies 21 , it is a cancellative pocrim. We require the following technical lemma.

LEMMA 8.10. Let a, b g Ak with a ) b and let 1 F n g ␻.

Ž. Ž. AkŽ. aSuppose b G 2k with a eën and b odd or ïce ërsa . If j2 n b, a - A k Ž. AkŽ. jb2 ny12,a then jn b, a s b y 2n y 1 and a y b - 2k y 2n. Ž. AkŽ. AkŽ. bIn all other cases, ja00,bsjb,a. Ž. AkŽ kk . k AkŽkk. cIf k G 5, then jmm2,2 y1s2 ymy1and j 2 y 1, 2 s k 2ymy2, for m sy1, 0, 1, . . . , 2k y 1. Proof. Ž.a and Ž. c are proved by induction on n and m, respectively, whileŽ. b is easily verified. We omit the tedious details.

OROLLARY A ␻ A C 8.11. k g J2 kkfor all k g , and f J2ky1when k G 5. Thus the sequence C l Jn, n G 9, is a strictly ascending chain of ¨arieties of cancellati¨e pocrims.

Proof. Suppose A k does not satisfy Ž.J2 k . Byw BR95, Lemma 17Ž. ix , it A k Ž. Ak Ž. follows that there exist a, b g Ak such that jb2 k ,a-jb2 ky1 ,a. Clearly a / b and, by symmetry, we may assume that a ) b. However, Lemma Ž.Ž. 8.10 a , b show that this implies a y b - 0, a contradiction. Thus A k g k Ž. Ž. J2 k .If kG5 then 2 y 2k y 1 y 2 G 2n so Lemma 8.10 c shows that A k f J2 ky1.

We now construct a cancellative pocrim A, into which each A k is embeddable, such that a subalgebra B of A has a homomorphic image which is not a pocrim. To do this, consider chains Ei, i g ␻, defined as follows:

E0000sÄ40,2,4,...,

E11sÄ4...,y4,y2,0,1,2,...,1111 and

EiisÄ4...,y2,y1,0,1,2,...iiii , i)1,

ϱ where, for each i, niii- m in E iff n - m. Let A s D is0Eiand define a linear order F on A by specifying that F restricts to the existing order Ž on each Eii, while for distinct i, j g ␻, n - mjiff i - j. In other words, as ²: ² : . a poset, A; F is the ordinal sum of the Ei; F , i g ␻. Define a 324 BLOK AND RAFTERY binary operation [ on A by

nij[ m s Ž.n q m iqjs m ji[ n .

The commutative integral pomonoid ²:A; [ ,0,F is residuated as follows: for integers n, m and i, j g ␻, we have nijy˙ m s 00 whenever nijFm, while for n ij) m , we have

Ž.nm,ifnmis even, or i j ) 1, ¡yiyj y y nijy˙ m s~ or i y j s 1 and n y m G 0, ¢Ž.nymq1iyj , otherwise.

Thus, A s ²:A; [ , y˙ , 0 is a pocrim. Notice that A is cancellative. Fact 8.12. A is isomorphic to a subalgebra of an ultraproduct of the pocrims A k , k g ␻. The lengthy but straightforward proof of this fact is given inwx BR93 , for a sequence of algebras very similar to our A k , k g ␻, and will therefore be omitted here. As a matter of interest, it can be checked quite easily that A has exactly one nontrivial congruence, viz., the congruence ␸I , where I is Ž. Ž. the unique nontrivial ideal E0 of A. Thus HA :C, and by Proposition 8.5, HAŽ y. :BCK. B A ϱ o Let be the subalgebra of with universe B s D is02E i. Write E2iand e E2 i, respectively, for the sets Ä4Ä4n2 i: n odd and n2 i: n even . Consider the o relation ␩ on B defined by: a␩ b iff a, b g E02,ora,bgE ifor some i,or e a,bgE2 i for some i. It is readily verified that ␩ is a congruence of B Ž y. hence also of B with 0r␩ s E0. Note that ␩ is not a BCK-congruence y of B , since 1220022r␩ y˙˙0 r␩ s 2 r␩ s 0 r␩ s 0 r␩ y 1 r␩ but 1 2r␩ / Ž. Žy. 02r␩. Thus, HS A M and HS A BCK. Each of the pocrims A k , however, belongs to one of the varieties Jn l C of cancellative pocrims. ŽÄ4. Since B g SPu A k : k g ␻ has a homomorphic image Br␩ which is not a pocrim, B cannot belong to any variety ofŽ. cancellative pocrims, whence we conclude:

THEOREM 8.13. There is no largest ¨ariety of cancellati¨e pocrims. Notice that the above argument also gives an alternative proof of Proposition 8.3. Inwxwx WK84 and BR93 , it was shown that there is no largest variety of BCK-algebras and no largest variety of pocrims. The method of proof inwx BR93 is similar to the argument above but used noncancellativeŽ. in fact, finite algebras in place of our A k . The essential difference between the two arguments lies in ourŽ. seemingly unavoidable use of the varieties Jn l C here. VARIETIES OF RESIDUATED POMONOIDS 325

The above construction shows that given aŽ. cancellative pocrim A, the condition HAŽ.:Cdoes not imply the condition HSŽ. A : C. It is also the case that HSŽ. A : C does not imply HSPŽ. A : C. Consider the same sequence of cancellative pocrims A k , k g ␻. By the remarks preceding Theorem 8.13, no variety ofŽ. cancellative pocrims contains all of the A k . Let D be the ‘‘direct sum’’ of these algebras, i.e., the subalgebra of Ł kkA whose elements are just those of Ł kkA which are zero in all but finitely Ž. Ž. many co-ordinates k. Since each Ak is in IS D , we clearly have HSP D C. To conclude that HSŽ. D : C, we need only check that homomorphic images of 2-generated subalgebras of D satisfy the quasi-identityŽ. 5 . This is indeed so: every finitely generated subalgebra of D is a subalgebra of the Ä4 direct product E of some finite subfamily of A k : k g ␻ and by Corollary 8.11, any such finite product E lies in the variety C l Jn for some n g ␻, Ž. Ž. Ž. so that HS E : C l Jn. In particular, any algebra in HS E satisfies 5 , as required. These distinctions were already drawn for the quasivarieties M and BCK inwxw RS92 and BR93 x .

9. PROBLEMS Problem 1. Is the poset P¨ Ž.M of all varieties of pocrims a lattice? In other words, is the union of any pair of pocrim varieties always contained in a pocrim variety? Is the lattice L¨ Ž.C of varieties of cancellative pocrims a chain? At this stage, our only knowledge of L¨ Ž.C is that it contains the infinite chain

O $ ISPPuŽ.␻ s J012l C : J l C : J l C : иии , where O is the trivial variety. How many varieties of cancellative pocrims are there?

¨ Problem 2. Are G and ISPPuŽ.␻ the only atoms of the poset P Ž.M ? Problem 3. Is every variety of pocrims congruence 3-permutable? If not, is there a bound on the degrees of permutability of pocrim varieties? Problem 4. Let V be a variety of pocrims. Is SŽV y. necessarily a variety? In other words, is HSŽV y.Ž: BCK? If so, then the first question of Problem 3 has an affirmative answer, by Theorem 3.3.. In general, we do not even know whether HŽV y. : BCK.If V is a variety of cancellative pocrims then HŽV y. : BCK, by Proposition 8.5, but we do not know whether HSŽV y. : BCK. Problem 5. If K is a variety of BCK-algebras with an equational base ⌺ then the class of all pocrims which are models of ⌺ is a variety V with SŽVy.Ž:Kthis inclusion being sharp in some cases.Ž , so S V y.is a variety of BCK-algebras. 326 BLOK AND RAFTERY

Under what conditions does the equality SŽV y. s K obtain? In particular, for integers n ) 1, is every BCK-algebra in En a residuation subreduct of an Ž.n q 1 -potent pocrim? The answer is affirmative if the BCK-algebra sq is in En , by Theorem 6.9.

sq Problem 6. Are there integers n ) 2 for which the variety En Žor, s equivalently, En . is locally finite?

ACKNOWLEDGMENT

The second author thanks the Department of Mathematics, Statistics and Computer Science of the University of Illinois at Chicago for its hospitality during the fall semesters of 1991 and 1994, when most of the work described in this paper was done.

REFERENCES

wxAme82 K. Amer, ‘‘Commutative monoids with monus,’’ Ph.D. thesis, University of Water- loo, 1982. wxAme84 K. Amer, Equationally complete classes of commutative monoids with monus, Algebra Uniërsalis 18 Ž.1984 , 129᎐131. wxBir84 G. Birkhoff, ‘‘Lattice Theory,’’ Colloquium Publications, Vol. 25, 3rd ed., reprinted with corrections, Am. Math. Soc., Providence RI, 1984. wxBF93 W. J. Blok and I. M. A. Ferreirim, Hoops and their implicational reducts Ž.Abstract , in ‘‘Algebraic Methods in Logic and in Computer Science,’’ Banach Center Publications, Vol. 28, Institute of Mathematics, Polish Academy of Sci- ences, Warsaw, 1993. wxBF W. J. Blok and I. M. A. Ferreirim, On the variety of hoops, preprint, 1996. wxBKP84 W. J. Blok, P. Kohler,¨ and D. Pigozzi, On the structure of varieties with equationally definable principal congruences II, Algebra Uniërsalis 18 Ž.1984 , 334᎐379. wxBP82 W. J. Blok and D. Pigozzi, On the structure of varieties with equationally definable principal congruences I, Algebra Uniërsalis 15 Ž.1982 , 195᎐227. wxBP88 W. J. Blok and D. Pigozzi, Local deduction theorems in algebraic logic, ‘‘Alge- braic Logic,’’ Colloquia Mathematica Societatis Janos Bolyai, Vol. 54. Budapest Ž.Hungary , 1988, 75᎐109. wxBP89 W. J. Blok and D. Pigozzi, ‘‘Algebraizable Logics,’’ Memoirs of the American Mathematical Society, Vol. 77, No. 396, Am. Math. Soc., Providence, 1989. wxBP94 W. J. Blok and D. Pigozzi, On the structure of varieties with equationally definable principal congruences III, Algebra Uniërsalis 32 Ž.1994 , 545᎐608. wxBR93 W. J. Blok and J. G. Raftery, Failure of the congruence extension property in BCK-algebras and related structures, Math. Japonica 38 Ž.1993 , 633᎐638. wxBR95 W. J. Blok and J. G. Raftery, On the quasivariety of BCK-algebras and its subvarieties, Algebra Uniërsalis 33 Ž.1995 , 68᎐90. wxBos69 B. Bosbach, Komplementare¨ Halbgruppen. Axiomatik und Arithmetik, Fund. Math. 64 Ž.1969 , 257᎐287. VARIETIES OF RESIDUATED POMONOIDS 327 wxBS81 S. Burris and H. P. Sankappanavar, ‘‘A Course in Universal Algebra,’’ Graduate Texts in Mathematics, Vol. 78. Springer-Verlag, New York, 1981. wxCha81 I. Chajda, Distributivity and modularity of lattices of tolerance relations, Algebra Uniërsalis 12 Ž.1981 , 247᎐255. wxCha88 I. Chajda, Tolerances in congruence permutable algebras, Czechosloäk Math. J. 38 Ž.1988 , 218᎐225. wxCha58 C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 Ž.1958 , 467᎐490. wxCor81 W. H. Cornish, A large variety of BCK-algebras, Math. Japonica 26 Ž.1981 , 339᎐344. wxCor82a W. H. Cornish, On Iseki’s´ BCK-algebras, in Lecture Notes in Pure and Applied Mathematics, Vol. 74, pp. 101᎐122, Dekker, New York, 1982. wxCor82b W. H. Cornish, BCK-algebras with a supremum, Math. Japonica 27 Ž.1982 , 63᎐73. wxCor84 W. H. Cornish, BCK-algebras with a supremum II. Distributivity and interpola- tion, Math. Japonica 29 Ž.1984 , 439᎐447. wxDie66 A. Diego, ‘‘Sur les Algebres` de Hilbert,’’ Collection de Logique Mathematique, Ser. A, No. 21, Gauthier-Villars, Paris, 1966. wxDyr87 K. Dyrda, None of the variety En, n G 2, is locally finite, Demonstratio Math. 20 Ž.1987 , 215᎐219. wxEva69 T. Evans, Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. 1 Ž.1969 , 399᎐403. wxFer92 I. M. A. Ferreirim, ‘‘On Varieties and Quasivarieties of Hoops and Their Reducts,’’ Ph.D. thesis, University of Illinois at Chicago, 1992. wxFle88 I. Fleischer, Every BCK-algebra is a set of residuables in an integral pomonoid, J.Algebra 119 Ž.1988 , 360᎐365. wxGra79¨¨ G. Gratzer, ‘‘Universal Algebra,’’ 2nd ed., Springer-Verlag, New York, 1979. wxHag73 J. Hagemann, ‘‘On Regular and Weakly Regular Congruences,’’ Preprint No. 75, Technische Hochschule Darmstadt, June 1973. wxHH79 J. Hagemann and C. Herrmann, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. ()Basel 32 Ž.1979 , 234᎐245. wxHig84 D. Higgs, Dually residuated commutative monoids with identity element as least element do not form an equational class, Math. Japonica 29 Ž.1984 , 69᎐75. wxIdz83 P. M. Idziak, On varieties of BCK-algebras, Math. Japonica 28 Ž.1983 , 157᎐162. wxIse66´´ K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 Ž.1966 , 26᎐29. wxIse79´´ K. Iseki, BCK-algebras with conditionŽ. S , Math. Japonica 24 Ž1979 . , 107᎐119. wxIse80´´ K. Iseki, On BCK-algebras with conditionŽ. S , Math. Japonica 24 Ž1980 . , 625᎐626. wxIT78 K. Iseki´ and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23 Ž.1978 , 1᎐26. wxKal60 J. A. Kalman, Equational completeness and families of sets closed under subtrac- tion, Nederl. Acad. Wetensch. Proc. Ser. A 63 Ž.1960 , 402᎐406. wxKea93 K. Kearnes, Congruence permutable and congruence 3-permutable locally finite varieties, J. Algebra 156 Ž.1993 , 36᎐49. wxKoh81¨¨ P. Kohler, Brouwerian semilattices, Trans. Amer. Math. Soc. 268 Ž.1981 , 103᎐126. wxKru24 W. Krull, Axiomatische Begruendung der allgemeinen Ideal Theorie, Sitzung der Physikalisch-Medicinische Societaet zu Erlangen 56 Ž.1924 , 47᎐63. wxMMT87 R. N. McKenzie, G. F. McNulty, and W. F. Taylor, ‘‘Algebras, Lattices, Varieties,’’ Vol. 1, Wadsworth & BrooksrCole, Monterey, CA, 1987. wxMit71 A. Mitschke, Implication algebras are 3-permutable and 3-distributive, Algebra Uniërsalis 1 Ž.1971 , 182᎐186. 328 BLOK AND RAFTERY wxMon60 A. Monteiro, ‘‘Cours sur les Algebres` de Hilbert et de Tarski,’’ Instituto Mat. Univ. del Sur, Bahia Blanca, 1960. wxMun86 D. Mundici, MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica 31 Ž.1986 , 889᎐894. wxOK85 H. Ono and Y. Komori, Logics without the contraction rule, J. Symbolic Logic 50 Ž.1985 , 169᎐201. wxOwe74 T. M. Owens, ‘‘Varieties of Skolem Rings,’’ Ph.D. thesis, Purdue University, 1974. wxPał80 M. Pałasinski, Some remarks on BCK-algebras, Math. Seminar Notes Kobe Uni¨. 8 Ž.1980 , 137᎐144. wxPał81 M. Pałasinski, On ideal and congruence lattices of BCK-algebras, Math. Japonica 26 Ž.1981 , 543᎐544. wxPał82 M. Pałasinski, An embedding theorem for BCK-algebras, Math. Seminar Notes Kobe Uni¨. 10 Ž.1982 , 749᎐751. wxPała M. Pałasinski,´ Subdirectly irreducibles in En-varieties of BCK-algebras, preprint. wxPałb M. Pałasinski,´ BCK-algebras and TD terms, preprint. wxPon86 B. Pondelıcek,ˇ´ˇ On a certain class of BCK-algebras with conditionŽ. S , Math. Japonica 31 Ž.1986 , 775᎐782. wxRRS91 J. G. Raftery, I. G. Rosenberg, and T. Sturm, Tolerance relations and BCK-algebras, Math. Japonica 36 Ž.1991 , 399᎐410. wxRS92 J. G. Raftery and T. Sturm, Tolerance numbers, congruence n-permutability and BCK-algebras, Czechosloäk Math. J. 42 Ž.1992 , 727᎐740. wxStu82 T. Sturm, On commutative BCK-algebras embeddable into directed commutative BCK-algebras, Math. Japonica 27 Ž.1982 , 197᎐212. wxStu T. Sturm, personal communication. wxTan75 S. Tanaka, On n-commutative algebras, Math. Seminar Notes Kobe Uni¨. 3 Ž.1975 , 59᎐64. wxTra79 T. Traczyk, On the variety of bounded commutative BCK-algebras, Math. Japon- ica 24 Ž.1979 , 283᎐292. wxTra88 T. Traczyk, On the structure of BCK-algebras with zx и yx s zy и xy, Math. Japon- ica 33 Ž.1988 , 319᎐324. wxWD39 M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 Ž.1939 , 335᎐354. wxWro83 A. Wronski,´ BCK-algebras do not form a variety, Math. Japonica 28 Ž.1983 , 211᎐213. wxWro85 A. Wronski,´ An algebraic motivation for BCK-algebras, Math. Japonica 30 Ž.1985 , 187᎐193. wxWK84 A. Wronski´´ and J. K. Kabzinski, There is no largest variety of BCK-algebras, Math. Japonica 29 Ž.1984 , 545᎐549.