Quasi-identities of finite semigroups and symbolic dynamics (original) (raw)

The Universality of the variety of quasigroups

Journal of the Australian Mathematical Society, 1976

The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra A,/ o ,/,,/ 2 , • • •) where f o ,fi,fi, • • • is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup (B, • > and polynomial operations F o , F,, F 2 , • • • over (B, •) such that (A, /",/,,•••) is a subalgebra of (B, F o , F,, • • •) satisfying exactly the same identities. Moreover, if there are only finitely many / 0 ,/i, • • •, then (B, •) can be taken so that its identities are recursive in those of (A,f o ,f,, • • •). If (A,f o ,f,, • ••> is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.

On Certain Quasivarieties of Quasi-MV Algebras

Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of sub-varieties of quasi-MV algebras has already been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its apparent logical repercussions, we tackle the issue in the present paper. We especially focus on quasivarieties whose generators either are subalgebras of the standard square quasi-MV algebra S, or can be obtained therefrom through the addition of some fixpoints for the inverse.

A survey on varieties generated by small semigroups and a companion website

Journal of Algebra

The aim of this paper is to provide an atlas of identity bases for varieties generated by small semigroups and groups. To help the working mathematician easily find information, we provide a companion website that runs in the background automated reasoning tools, finite model builders, and GAP, so that the user has an automatic intelligent guide on the literature. This paper is mainly a survey of what is known about identity bases for semigroups or groups of small orders, and we also mend some gaps left unresolved by previous authors. For instance, we provide the first complete and justified list of identity bases for the varieties generated by a semigroup of order up to 4, and the website contains the list of varieties generated by a semigroup of order up to 5. The website also provides identity bases for several types of semigroups or groups, such as bands, commutative groups, and metabelian groups. On the inherently non-finitely based finite semigroups side, the website can decide if a given finite semigroup possesses this property or not. We provide some other functionalities such as a tool that outputs the multiplication table of a semigroup given by a C-presentation, where C is any class of algebras defined by a set of first order formulas. The companion website can be found here http://sgv.pythonanywhere.com

On finite hyperidentity bases for varieties of semigroups

Algebra Universalis, 1995

For any variety V of semigroups, we denote by H(V) the collection of all hyperidentities satisfied by V. It is natural to ask whether, for a given V, H(V) is finitely based. This question has so far been answered, in the negative, for four varieties of semigroups: for the varieties of rectangular bands and of zero semigroups by the author in [8]; for the variety of semilattices by Penner in [5]; and for the variety S of all semigroups by Bergman in [1]. In this paper, we show how Bergman's proof may in fact be used to deal with a large class of subvarieties of S, namely all semigroup varieties except those satisfying x 2= x 2+m for some m. As a first step in the investigation of these exceptional varieties, we also present some hyperidentities satisfied by the variety B1.1 of bands, and, using the same technique, show that H(V) is not finitely based for any subvariety V of B~a. These proofs all exploit the fact that the particular variety in question has hyperidentities of arbitrarily large arity. We conclude with an example of a variety for which even the collection of hyperidentities containing only one binary operation symbol is not finitely based.

Complexity of Some Problems Concerning Varieties and Quasi-Varieties of Algebras

SIAM Journal on Computing, 2000

In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasi-identities? (3) Do A and B have the same set of term operations?

Semi-lattice of varieties of quasigroups with linearity

Algebra and Discrete Mathematics, 2021

A σ-parastrophe of a class of quasigroups A is a class σA of all σ-parastrophes of quasigroups from A. A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch. A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are pres...

About quasivarieties of p-algebras and Wajsberg algebras

for the financial support, which took the form of teaching assistantship during the regular terms and of research assistantship during the summers, of the last four and a half years. I also wish to show here my gratitude to the " Universidad de los Andes" in Venezuela for helping me financially in my graduate studies. This work is affectionally dedicated to my son Daniel Guillermo and to my wife Magdalena. we found an example of a proper quasivariety of p-algebras which generates the whole variety. In constructing such a quasivariety, we took full advantage of the. results obtained by Adams in [1]. We will present this example in the last section of this chapter. Basic Concepts of Universal Algebra An n-ary operation / on a set A is any function / : A**-* A. An algebra is an ordered pair (A] F) where A is a set and F is a set of operations on A. To each f € F corresponds a non-negative integer, namely: its arity. We will consider only algebras for which F is finite. The elements of F are called the basic operations of the algebra and the set A, its universe or carrier set. If F = {/j,..., /j^}, we often write {A] fi,ffg). If ni,... are the corresponding arities of the elements in F we say that A is an algebra of type (n^,...usually adopting the convention > • ' • > nj^. ]i {A', F) and (i4'; F') are algebras of the same type, there is a bijective correspondence between F and F' such that if /' € F' corresponds to / € F, both / and /' have the same arity. Since we always consider only algebras of the same type, we will use the same symbol for a given basic operation in all the algebras under consideration. Also, we will write A instead of {A; F) when this causes no confusion. There are three fundamental methods of constructing new algebras: the for mation of subalgebras, homomorphic images and direct products. If A and B are algebras of the same type, A is a subalgebra of By in symbols, A < B, if A Ç B and every basic operation of A is the restriction of the corresponding basic operation of B. A homomorphism from A to B is a function a : A-> B such that for any

Semilattice sums of algebras and Mal’tsev products of varieties

Algebra universalis, 2020

The Mal'tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal'tsev product V • S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal'tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V • S. Let K be a quasivariety of Ω-algebras, and Q and R, two of its subquasivarieties. Assume additionally that R-algebras are idempotent. Then the Mal'tsev product Q • K R of Q and R relative to K consists of K-algebras A with a congruence θ such that A/θ is in R, and each θ-class a/θ is in Q. Note that by the idempotency of R, each θ-class is always a subalgebra of A. If K is the variety of all Ω-algebras, then the Mal'tsev product Q • K R is called simply the Mal'tsev product of Q and R, and is denoted by Q • R. It follows by Mal'tsev results [15] that Mal'tsev product Q • K R is a quasivariety. In this paper we are interested in Mal'tsev products V • S such that V is a variety of Ω-algebras and S is the variety of the same type as V, equivalent to the variety of semilattices. Members of V • S are disjoint unions of V-algebras over its semilattice homomorphic image, and are known as semilattice sums of V-algebras. The product V • S is a quasivariety. However, until recently it was not known under what conditions it is a variety. The main result of this paper (Theorem 5.3) shows that if V is a strongly irregular variety of a type with no nullary Key words and phrases. Mal'tsev product of varieties, semilattice sums, P lonka sums, Lallement sums, regular and irregular identities, regularization and pseudoregularization of a variety.