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 mo- des, 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 commu- tative 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 commu- tative 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. Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw University of Technology 2021-09-29 322 K. Matczak and A. B. Romanowska 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 pa- per 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 equiv- alent 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. Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw University of Technology 2021-09-29 Quasivarietiesof...BinaryModes 323 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 vari- eties of commutative binary modes, i.e. modes with a binary commutative multiplication. These varieties are equivalent to certain varieties of com- mutative quasigroup modes (quasigroups with commutative multiplication Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw University of Technology 2021-09-29 324 K. Matczak and A. B. Romanowska 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) = ∞. Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw University of Technology 2021-09-29 Quasivarietiesof...BinaryModes 325 (b) If f(∞)=0,thenMR is a variety and is defined by the unique identity f f(i) x pi =0.