Quasivarieties of Cancellative Commutative Binary Modes

K. Matczak Quasivarieties of Cancellative A. B. Romanowska Commutative Binary Modes

Abstract. The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive. Keywords: quasivarieties, modes, subreducts, aﬃne spaces, commutative quasigroup modes, commutative binary modes.

1. Introduction

Commutative binary modes (or CBM-groupoids) were intensively investi- gated by Jeˇzek and Kepka (see [8], [9], [10], [11], [12]) under the name of commutative idempotent medial groupoids, and by Romanowska and Smith (see [18], [19], [21]). In particular, the lattice of varieties of such groupoids was first described in [9], and a structural characterisation was provided in [19]. The lattice of varieties splits into two parts. The irregular varieties are known to be equivalent to varieties of commutative quasigroup modes and to varieties of affine spaces over the rings Z2k+1. (See [18] and [21].) The remaining varieties are regular, and with the exception of the variety of all CBM-groupoids, they are regularisations of irregular ones. Basic facts on modes, commutative binary modes, commutative quasigroup modes and affine spaces that we need in this paper are recalled briefly in Section 2. For more information, we refer the reader to the monographs [18] and [21]. Until recently, the only known facts concerning quasivarieties of commutative quasigroup modes and commutative binary modes were provided by Hogben and Bergman [6] who characterised all quasivarieties of commutative quasigroup modes that are varieties. See also [2] and [3] for related results. The main goal of this paper is to initiate investigations that should even- tually lead to a characterisation of the lattice of all quasivarieties of commutative binary modes, and more generally to the lattices of quasivarieties of some other classes of modes. While this is a rather long-term project, some progress has been made on an initial stage.

Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko; Received September 20, 2002, Accepted February 15, 2003

Studia Logica 78: 321–335, 2004. c 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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In this paper we investigate the quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Further results on the quasivarieties of commutative binary modes will be provided in another paper presently under preparation. We start with a description of the lattice of quasivarieties of affine spaces over a principal ideal domain. This was possi- ble due to the characterisation of the lattice of quasivarieties of modules over such a ring given by D. V. Belkin in his doctoral dissertation [1]. The two lattices are isomorphic. Since the variety of commutative quasigroup modes is equivalent to the variety of affine spaces over the principal ideal domain D of rational dyadic numbers, and the lattices of subquasivarieties of equivalent quasivarieties are isomorphic, this immediately gives us a description of the lattice of quasivarieties of commutative quasigroup modes. We also provide an easy algorithm for translating quasi-equational bases from one of the languages used to the other. This is discussed in Section 3 and Section 4. The relatively straightforward results of Section 4 are necessary to under- stand and prove the main results of the next Section. Section 5 brings us to quasivarieties of cancellative commutative binary modes. On the one hand it is very well known that each such mode embeds as a subreduct into an affine space over D (or equivalently into a commutative quasigroup mode). (See e.g. [20] and [12].) On the other hand, the class of subreducts (of given type) of the algebras in a given quasivariety is again a quasivariety ([13], Section V.11.1). These two facts imply that each quasivariety of affine spaces over D determines a quasivariety of cancellative binary modes. We show that these quasivarieties are precisely all the quasivarieties of cancellative commutative binary modes. We provide quasi-equational bases for them. They are also all quasivarieties of commutative binary modes that are embeddable into affine spaces. It is worth noting that, in many aspects, the commutative binary modes behave like barycentric algebras over D instead of R. (See [18] and [21].) The essential differences concern the lattices of varieties and, as we will see, the lattices of subquasivarieties. (The lattice of varieties of barycentric algebras was described by Neumann [15], and the lattice of quasivarieties by Ignatov [7].) In particular, there is only one quasivariety of cancellative barycentric algebras (the quasivariety of convex real sets), but there are uncountably many quasivarieties of cancellative commutative binary modes. For algebraic concepts and notations used in this paper, the readers are referred to [21] and [22]. In particular, mappings are generally placed in the natural position on the right of their arguments. These conventions help to minimise the number of brackets, which otherwise proliferate in the study of non-associative systems such as modes and affine spaces.

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2. Modes

Most algebras considered in this paper are modes in the sense of [18] and [21], algebras in which each element forms a singleton subalgebra, and for which each operation is a homomorphism. For algebras (A, Ω) of a given type τ :Ω→ N, these two properties are equivalent to satisfaction of the identity x...xψ = x (1) of idempotence for each operation ψ in Ω, and the identity

(x11 ...x1mψ) ...(xn1 ...xnmψ)φ =(x11 ...xn1φ) ...(x1m ...xnmφ)ψ (2) of entropicity for any two operations (m-ary) ψ and (n-ary) φ in Ω. One of the main families of examples of modes is given by affine spaces over a commutative ring with unity, or, more generally, by subreducts (sub- algebras of reducts) of affine spaces. Affine spaces are considered here as modes in the following way (cf. [18] and [21]). Let R be a commutative ring with 1. An affine space over a commutative ring R with 1 (or an affine R-space) may be defined as the reduct (A, P, R) of an R-module (A, +,R), where P is the Mal’cev operation 3 P : A → A;(x1,x2,x3) → x1x2x3P = x1 − x2 + x3,

and R is the family of binary operations 2 r : A → A;(x1,x2) → x1x2r = x1(1 − r)+x2r

for each r ∈ R. Equivalently, it is defined as the full idempotent reduct of such a module. The class of all affine R-spaces is a variety. (See [4].) Equivalently, this variety may be defined as the class R of Mal’cev modes (A, P, R) with a ternary Mal’cev operation P and binary operations r for each r ∈ R satisfying certain identities given in [18]. By choosing an ar- bitrary element of a non-empty R-algebra (A, P, R) to be a zero, one may define the structure of an R-module on the set A. (See e.g. [21], Chapter 6.) The structure of a ring R is defined on the free R-algebra on two generators. We will identify these two equivalent descriptions of affine R-spaces. Many known varieties of groupoid modes are equivalent to varieties of affine spaces. (An equivalence of varieties of algebras is understood here as defined in [14], [21] and [5].) This in particular concerns the irregular varieties of commutative binary modes, i.e. modes with a binary commutative multiplication. These varieties are equivalent to certain varieties of commutative quasigroup modes (quasigroups with commutative multiplication

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that at the same time are modes). Now each variety of quasigroup modes, as a Mal’cev variety, is equivalent to a variety of aﬃne spaces. (See [21], Chapter 6.)

3. Quasivarieties of aﬃne spaces

In this section R will always denote a principal ideal domain. Our aim is to describe the lattice Lq(R)ofquasivarietiesofaﬃneR-spaces, and in particular the lattice Lq(D)ofquasivarietiesofaﬃneD-spaces. We start with Belkin’s description of the quasivarieties of R-modules. For a cardinal α =0,let K(α) be the set of functions

f : α ∪{∞}→ω ∪{∞},

where f(∞) ∈{0, ∞} and f(∞) = 0 implies that f(α) ⊆ ω and f(i)=0 for almost all i ∈ α.ThenK(α) is a distributive lattice with respect to the following operations:

(f ∨ g)(i)=max{f(i),g(i)}, (f ∧ g)(i)=min{f(i),g(i)},

where i<∞ for all i ∈ ω.LetP (R)bethesetofrepresentativesof irreducible (and hence prime) elements of R modulo invertible elements, ordered into the type of α. Then the following holds.

Theorem 3.1. [1] Let R be a principal ideal domain and let |P (R)| = α. Then the lattice L (Mod ) of quasivarieties of R-modules is isomorphic to q R the lattice K(α), L (Mod ) ∼ K(α). q R = The isomorphism is given by

ϕ : K(α) →L (Mod ); f → MR , q R f

where the subquasivarieties MR are deﬁned as follows. f

(a) If f(∞)=∞,thenMR is deﬁned by the quasi-identities f

f(i)+1 → f(i) (xpi =0) (xpi =0) (3)

for pi in P (R) and f(i) = ∞.

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(b) If f(∞)=0,thenMR is a variety and is deﬁned by the unique identity f f(i) x pi =0. (4) f(i)=0

In particular, Belkin’s Theorem recovers the description of the lattice of quasivarieties of abelian groups given earlier by Vinogradov [23]. The quasivarieties of R-modules are also determined by certain sets of finitely generated R-modules. The structure of finitely generated R-modules is very well known (see e.g. [22], Section II.3.3), and carries over to affine R-spaces. Each such non-zero module is a direct product of a free module and a torsion module. It is well-known that a free R-module (XModR, +,R) over a set X is the direct sum x∈X Rx,whereeachRx is isomorphic to the R-module R. This direct sum is isomorphic to the direct power Rn of R in the case where the set X is finite and consists of n elements. Then the affine R-space (XModR,P,R) is the free affine R-space over the set {0}∪X.(See [4] and [21].) The torsion module has a uniquely defined decomposition as the direct product of modules E(p) for all p in P (R) such that E(p) =0: E(p)=R/(pn1 ) ⊕ ...⊕ R/(pnj ),

where 1 ≤ n1 ... ≤ nj.NowletH(R) be the set consisting of the module R and the cyclic modules R/(pni )forp ∈ P (R). Proposition 3.2. Two quasivarieties Q and Q of R-modules are equal if 1 2 and only if Q ∩ H(R)=Q ∩ H(R), 1 2 so the quasivarieties of R-modules are determined by the members of the set H(R) they contain. Proof. The proof follows by the structure of ﬁnitely generated R-modules described above. The details are similar to the case of abelian groups. (See [23].)

The next proposition shows in detail which members of H(R)agiven quasivariety MR contains. f Proposition 3.3. For a quasivariety MR of R-modules the following hold: f (a) If i = ∞ and f(i)=m<∞,thenMR contains the modules R/(pn) f i for n ≤ m but not R/(pm+1).Iff(i)=∞,thenMR contains the i f m modules R/(pi ) for all m.

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(b) If f(∞)=∞,thenMR contains the module R.Iff(∞)=0,then f MR does not contain the module R, and contains a finite number of f n modules R/(pi ). Proof. It is sufficient to note that the free module R does not satisfy any of the identities (4), but does satisfy all the quasi-identities (3). One may then check which quasi-identities of Belkin’s Theorem are satisfied by members of H(R) different from R.

The lattices of varieties of R-modules and of affine R-spaces are known to be isomorphic. (See [4] and [21].) Both are dually isomorphic to the lattice of ideals of the ring R.NowifaquasivarietyMR is a variety, then f the identity (4) defining it is equivalent to a certain affine space identity. To show this, note that the following identities are equivalent: f(i) x pi =0, ( )=0 f i f(i) f(i) x pi = y pi , ( )=0 ( )=0 f i f i − f(i) f(i) y( pi )+x( pi +1)=x. f(i)=0 f(i)=0 The last identity is in fact an affine space identity f(i) yx pi +1= x. (5) f(i)=0

It defines the corresponding variety R of affine R-spaces. f A similar “translation” may be made for quasivarieties. Lemma 3.4. Each quasi-identity (3) among those defining a quasivariety MR is equivalent to the affine space quasi-identity f f(i)+1 → f(i) (yxpi +1= x) (yxpi +1= x), (6)

where pi ∈ P (R) and f(i) = ∞. Proof. First note that if ak =0foranelementa = x − y of an R-module M,thenalsoyxk +1 = x. Hence the quasi-identity (3) is not satisﬁed in f(i)+1 M precisely if there are elements x and y in M such that yxpi +1= x f(i) and yxpi +1= x. This however means that the quasi-identity (6) is not satisﬁed.

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Corollary 3.5. Let R be a principal ideal domain, and let |P (R)| = α. Then the lattice Lq(R) of quasivarieties of aﬃne R-spaces is isomorphic to the lattice K(α). For any f in K(α), the corresponding subquasivariety R f of R is deﬁned by the quasi-identities (6) with pi ∈ P (R) and f(i) = ∞ in the case where f(∞)=∞, or by the unique identity (5) in the case f(∞)=0.

Proof. This follows directly by the equivalence of module and affine space identities discussed above, Lemma 3.4, and the structural characterization of finitely generated affine R-spaces.

In particular, we consider the quasivarieties of aﬃne D-spaces. The set D L D L P ( ) consists of odd prime numbers. The lattices q( ), q(ModD)and K(ω) are isomorphic. Each ideal of D has the form (2k + 1). So the sub- lattice of Lq(D) consisting of varieties is isomorphic to the lattice of odd positive integers under division with a greatest element added. Denote the subvarieties of D by D . (Note that 2k + 1 decomposes as a product of 2k+1 powers of odd primes.) Then the subvariety D is deﬁned by the identity 2k+1

x(2k +2)− y(2k +1)=x, (7)

or equivalently yx2k +2= x (8) corresponding to the module identity x(2k + 1) = 0. Note as well that since the rings D/(2k +1)and Z2k+1 are isomorphic, it follows that the subvariety D is equivalent to the variety of affine Z2 +1-spaces. These varieties are 2k+1 k the only so-called deductive quasivarieties of affine D-spaces, i.e. they do not contain subquasivarieties that are not varieties. (Cf. [2] and [6].) Other subquasivarieties of D are defined by the quasi-identities (6) with the pi as odd prime numbers.

4. Quasivarieties of commutative quasigroup modes

As a Mal’cev variety, each quasigroup mode variety is equivalent to a variety of affine spaces. Note that the lattices of subquasivarieties of equivalent quasivarieties are isomorphic. (See e.g. [1].) Equivalence of quasivarieties is defined here similarly as for varieties, see e.g. [16] and [5]. Hence the lattices of subquasivarieties of a variety of quasigroup modes and of subquasivarieties of the equivalent variety of affine spaces are isomorphic. Both are isomorphic with the lattice of subquasivarieties of the corresponding module

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variety. In this section we will describe the lattice Lq(CQM)ofquasivari- eties of commutative quasigroup modes (or CQM-quasigroups). For more information about such quasigroups, see e.g. [21]. First note that the variety CQM of commutative quasigroup modes is equivalent to the variety D of affine D-spaces. For an affine D-space A, the equivalent quasigroup is defined on the set A by taking the operations 2, 2−1 and −1 as the basic quasigroup operations. The lattice of subvarieties of CQM was first described in [9] (see also [18]), and is isomorphic to the lattice of odd natural numbers under division, with a greatest element added. The first characterization of the lattice of subquasivarieties of CQM follows easily from the above discussion.

Theorem 4.1. The lattices Lq(CQM) of quasivarieties of commutative qua- L D D L D sigroup modes, q( ) of aﬃne -spaces, q(ModD) of -modules and K(ω) are isomorphic. L ∼ L D ∼ L ∼ q(CQM) = q( ) = q(ModD) = K(ω).

Proof. The ﬁrst isomorphism follows by the equivalence of the varieties CQM and D and the fact that lattices of subquasivarieties of equivalent quasivarieties are isomorphic. The second isomorphism was discussed in the previous section.

In what follows, we will provide quasi-equational bases for each subquasivariety of CQM. As shown in [21, Chapter VI], each subvariety CQM 2k+1 of CQM is deﬁned by the identity

xyw2k+1 = x, (9)

where xyw2k+1 := y(zl−2(...(z1 · yx) ...)), (10)

with l = l(2k + 1) the least integer bigger than log2(2k +1)andeach zi in {x, y}.Letm =2k + 1. By setting x to be 0 and y to be 1 in the word wm, one obtains the binary form

wm(0, 1) = al(m)−1(al(m)−2(...(a1 · a00) ...))

−l(m) of the number m2 . Now a simple induction shows that the word xywm is interpreted in the class of 2−1-reducts of aﬃne D-spaces as

l(m) −l(m) wm =(my +(2 − m)x)2 .

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Hence the identity xywm = x can be interpreted as the aﬃne D-space identity yxm +1= x, (11) similarly as done with (9). Note that this identity deﬁnes the variety D , m and is also equivalent in the variety D to the following:

xym2−l(m) = x. (12)

Note more generally that if any of the equations (9), (11) and (12) holds for elements x and y of an affine D-space A, then the other two hold for these elements as well. Since all ideals of the ring D have the form (2k +1), ∼ and D/(2k +1)= Z2k+1, it follows that each subvariety CQM is in fact 2k+1 equivalent to the variety Z . Hence for f ∈ K(ω)withf(∞)=0,the 2k+1 corresponding quasivariety CQM coincides with CQM for some 2k+1, f 2k+1 it is a variety, and is equivalent to the variety of affine Z2k+1-spaces. It is s clear that the variety contains all Zps for the divisors p of 2k + 1, but not forhigherpowersofp. If f(∞)=∞, then the quasivariety CQM is defined by the quasi- f identities (xyw f(i)+1 = x) → (xyw f(i) = x) (13) pi pi

for odd prime numbers pi with f(i) = ∞. AsinthecaseofaﬃneD-spaces, the subquasivarieties of CQM are also determined by subsets of the set Hq(D) consisting of quasigroups equivalent to aﬃne space reducts of the modules in H(D).

5. Quasivarieties of cancellative commutative binary modes

A commutative binary mode or CBM-groupoid is cancellative if it satisﬁes the cancellation quasi-identity

(xy = xz) → (y = z).

The class CBM of cancellative CBM-groupoids is a subquasivariety of the cl variety CBM of commutative binary modes. Cancellative CBM-groupoids are known to be precisely the multiplicative subreducts of commutative quasigroup modes, or equvalently 2−1-subreducts of aﬃne D-spaces. (See [21, Sections 6.6 and 7.1].) Cancellative CBM-groupoids play a very special role in the theory of commutative binary modes. It is known that each such

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groupoid is a semilattice Lallement sum of cancellative subgroupoids over its semilattice replica, or equivalently it is a subgroupoid of a Plonka sum of cancellative subgroupoids over its semilattice replica. (See [17] and [21] for corresponding deﬁnitions and details.) In this section we will describe the quasivarieties of cancellative CBM-groupoids. First we formulate several of the facts concerning general lattices of quasivarieties and congruences of CBM-groupoids that we will need in the proof.

Proposition 5.1. [13, Section V.11.1] Let Q be a quasivariety of type τ1 : 1 Ω1 → N,andletQ be a quasivariety of type τ2 :Ω2 → N such that Ω1 ⊆ Ω2. 2 Then the class Q of Q -algebras that are embeddable as subreducts into Q - 1 2 algebras is itself a quasivariety.

Proposition 5.2. [13, Section VI.14.3]

(a) Each minimal variety and each minimal quasivariety is generated by its free algebra on two generators.

(b) Each minimal variety contains a unique minimal subquasivariety. There exist minimal quasivarieties not contained in any minimal variety.

We will also need some information about the congruences of the quasigroup D and of the CBM-groupoid D1 obtained as a subgroupoid D ∩ [0, 1] of the multiplicative reduct of the quasigroup D. Recall that the quasigroup D is a free algebra on two generators in the variety CQM, and the groupoid D1 is a free algebra on two generators in the variety CBM.

Lemma 5.3. [21, Section 6.6] Each non-trivial quasigroup congruence on the quasigroup D is of the form

σm := {(x, y) ∈ D × D | x − y ∈ mD}

for an odd natural number m. The congruence classes of σm are precisely the cosets of the ideal mD in the ring D. Moreover, ∼ D/mD = D/σm = Zm.

A congruence θ of a CBM-grupoid G is said to be cancellable if

ab θ ac =⇒ bθc.

Proposition 5.4. A non-trivial congruence θ on the CBM-groupoid D1 is cancellable if and only if there is an odd natural number m such that θ = σm.

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Proof. The proof goes like the proof of Proposition 453 in [18] character- ising fully invariant congruences of D1. (Note that the congruences σm are denoted by (m) there.) In fact the proof of that proposition describes all the congruences of D1, dividing them into two types denoted (m)and(m ). Only the congruences (m) are cancellable.

Let us also recall that every cancellative CBM-groupoidembedsasa subreduct into an affine D-space or equivalently into a commutative quasigroup mode. This follows by more general results of [20] and [12]. In fact, Theorem 5.3.1 of [12] shows that the embedding can be done in certain canonical way. Let Hg(D) denote the set of multiplicative reducts of quasigroups in the set Hq(D). They will be identified with the 2−1-subreducts of corresponding affine spaces. Theorem 5.5. The lattice L (CBM ) of quasivarieties of cancellative com- q cl mutative binary modes is isomorphic to the lattice Lq(CQM) of quasivarieties of commutative quasigroup modes. In particular, the following lattices are isomorphic: L (CBM ) ∼ L (CQM) ∼ L (D) ∼ L (Mod ) ∼ K(ω). q cl = q = q = q D = For each f in K(ω), the corresponding subquasivariety CBM of CBM is f cl defined by the same quasi-identities that define CQM as a subquasivariety f of CQM. Proof. In the proof, quasigroups are always from the class CQM and groupoids from the class CBM . We will identify them with the corre- cl sponding (sub)reducts of affine D-spaces. We start with the following observation. Note that by Theorem 5.1 each quasivariety CQM uniquely determines the quasivariety CBM of (mul- f f tiplicative) subreducts of quasigroups in CQM . Evidently, such a quasi- f variety also satisfies the quasi-identities defining the quasivariety CQM as f a subquasivariety of CQM, and contains precisely the members of Hg(D) that are multiplicative reducts of the members of Hq(D)thatbelongto CQM . In the case where the quasivariety CQM is a variety, the quasi- f f variety CBM is also a variety, and both are equivalent. (See [21, Section f 6.6].) For an algebra A,letQ(A) denote the quasivariety generated by A.We similarly denote the quasivariety generated by a set of algebras. Consider

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now the quasivariety Q(D1). First note that this quasivariety contains also the groupoid D. Indeed, this groupoid is a directed colimit of ﬁnitely generated subgroupoids, that in fact are two-generated, and each one is isomorphic to D1. It follows that a groupoid quasivariety contains the groupoid D1 if and only if it contains the groupoid D. Moreover the quasivariety Q(D1)=Q(D) is a minimal quasivariety. This follows by the facts that D1 is free on two generators, both in this quasivariety and in the variety CBM, and that the free algebra over a set X in a quasivariety and in the variety generated by this quasivariety coincide. It also follows that this quasivariety is the only minimal quasivariety that is not a variety. Now note that each two-generated cancellative CBM-groupoid is a quo- tient of the groupoid D1 by a cancellable congruence. By Proposition 5.4, such quotients are precisely groupoids D1 and Z2k+1 for all natural k.(Re- −1 call that in Z2k+1, the operation 2 coincides with k +1.) It follows that a non-trivial quasivariety Q of cancellative CBM-groupoids must contain a non-empty subset of the set Hg(D). Let H(Q)=Hg(D) ∩ Q. Such subsets H(Q) have properties similar to those of the subsets of H(D) determining D the subquasivarieties of .IfQ contains a groupoid Zpi ,thenitcontains ≤ D also all groupoids Zpj for j i.IfQ contains Zpi = /σpi for all i,then σpi =0, i∈ω D ≤ D ∈ where 0 is the equality relation. Hence i∈ω Zpi , implying that Q. There is a similar situation in the case where Q contains inﬁnitely many

Z ij ,wherej ∈ J := {j ∈ ω | f(j) =0 } .Then pj

σ ij =0, pj j∈J the equality relation again. Hence also in this case D ∈ Q. It follows that

Q contains either a finite number of Z ij (and in this case it may or may pj not contain D), or else it contains an infinite number of such groupoids and necessarily also D. We will show that Q(H(Q)) = Q. First note that in the case the set H(Q) is finite and does not contain D, the quasivariety Q(H(Q)) is a variety and is equivalent to the corresponding variety of CQM-quasigroups. Hence our claim holds. So assume now that H(Q)containsD.Byresultsof[12, Section 5.3] it is clear that each finitely generated Q-groupoid G embeds

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in a uniquely determined finitely generated CQM-quasigroup q(G). Such quasigroup q(G) is a direct product of some power Dn and a finite number of finite quasigroups from H(Q). The groupoid G embeds into this product as a subreduct. Now all finitely generated quasigroups q(G) obtained in this way from finitely generated CBM-groupoids generate a quasivariety of CQM-quasigroups. This quasivariety must coincide with some quasivariety CQM and is determined by the set of quasigroups in H(Q). On the other f hand, the quasivariety Q is determined by the groupoids in H(Q). It follows that indeed, Q = Q(H(Q)). Now let us define a function

h : L (CBM ) → K(ω); Q → f q cl as follows:

(a) If a quasivariety Q contains the groupoid D, then we put f(∞)=∞, otherwise f(∞)=0.

n (b) If Q contains the groupoids Zpi for a natural i, then we put f(i)= { | n ∈ } sup n Zpi H(Q) . It is easy to see that the functions f are indeed in K(ω), the mapping h is well deﬁned, it is bijective, and preserves the order. Hence it is a lattice isomorphism. Moreover, if h(Q)=f,then

H(Q)=H(CBMf ).

It follows by the ﬁrst part of the proof that the mapping g assigning to each quasivariety Q with h(Q)=f the quasivariety CBM is an order preserving f bijection, whence also a lattice isomorphism.

In a subsequent paper we will show how the quasivarieties of cancellative commutative binary modes determine quasivarieties of non-cancellative commutative binary modes that do not contain the variety of semilattices. In fact such quasivarieties are determined by a family of modes that are certain sums of groupoids belonging to the set Hg(D). The next step is to describe quasivarieties containing the variety of semilattices. Here the concept of a quasiregularisation of a quasivariety will play an essential role. (See [21, Chapter 4].) All these results depend on a good structural description of the algebras in question. Though we do have a good representation theorem for general

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commutative binary modes, it seems that investigation of their quasivarieties gives much deeper insight into their structure. The results of this paper are closely connected with two facts. The generator D1 of the variety CBM is a unit interval of the ordered ring D. On the other hand this mode is the free object on two free generators in the variety CBM. These two facts suggest that similar methods may be eﬀective in investigations of subquasivarieties of varieties of modes generated by a unit interval of other ordered subrings of the real ring and by a free subreduct on two free generators of a free aﬃne space over a principal ideal domain. Acknowledgments. The authors thank Kira Adaricheva for information on Russian publications concerning quasivarieties of modules, and Denis Belkin for providing the manuscript of his doctoral dissertation. Work on this paper was completed during the visit of the second author to Iowa State University, Ames Iowa, in Summer 2002.

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[12] Jeˇzek,J.,andT.Kepka, Medial Groupoids, Academia, Praha, 1983. [13] Mal’cev,A.I.,Algebraic Systems, Springer-Verlag, Berlin, 1973.

[14] McKenzie,R.N.,G.F.McNulty,andW.F.Taylor, Algebra, Lattices, Varieties, vol. I, Wadsworth, Monterey, 1987.

[15] Neumann, W. D., ‘On the quasivariety of convex subsets of aﬃne spaces’, Arch. Math. 21:11–16, 1970.

[16] Pigozzi, D., ‘A lattice characterization of equivalent quasivarieties’, Contemporary Mathematics (Part 3) 131:187–199, 1992.

[17] Plonka,J.,andA.Romanowska, ‘Semilattice sums’, in A. Romanowska, and J. D. H. Smith, (eds.), Universal Algebra and Quasigroup Theory, Helderman Verlag, Berlin, 1992, pp. 123–158.

[18] Romanowska,A.B.,andJ.D.H.Smith, Modal Theory, Heldermann, Berlin, 1985. [19] Romanowska,A.B.,andJ.D.H.Smith, ‘On the structure of semilattice sums’, Czechoslovak Math. J. 41:24–43, 1991. [20] Romanowska,A.B.,andJ.D.H.Smith, ‘Embedding sums of cancellatice modes into functorial sums of aﬃne spaces’, in J. M. Abe and S. Tanaka, (eds.), Unsolved Problems on Mathematics for the 21st Century, a Tribute to Kiyoshi Iseki’s 80th Birthday, 2001, IOS Press, Amsterdam, pp. 127–139. [21] Romanowska,A.B.,andJ.D.H.Smith, Modes, World Scientiﬁc, Singapore, 2002.

[22] Smith, J.D.H., and A.B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999. [23] Vinogradov, A. A., ‘Quasivarieties of abelian groups’, (in Russian), Algebra i Logika 4:15–19, 1965.

K. Matczak Faculty of Mathematics and Information Sciences Warsaw University of Technology Plac Politechniki 1 00 661 Warsaw, Poland matczak [email protected]

A. Romanowska Faculty of Mathematics and Information Sciences Warsaw University of Technology Plac Politechniki 1 00 661 Warsaw, Poland [email protected]

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