Varieties Of Commutative Semigroups Research Articles

Varieties of commutative semigroups closed under dominions

In this paper, we find sufficient conditions for some commutative varieties of semigroups to be closed under dominions.

The Ranks of the Planarity Varieties of Commutative Semigroups

We study the proposed L.M. Martynov, the rank of a planarity for varieties of commutative semigroups.

Endomorphism monoids in varieties of commutative semigroups

A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism.

Pre-strongly solid varieties of commutative semigroups

Pre-strongly solid varieties of commutative semigroups

All Reg-solid varieties of commutative semigroups

We determine all regular solid varieties of commutative semigroups. Each of them is contained in the Reg-hyperequational class V RC defined by the associative law and the commutative law, and every subvariety of V RC is regular solid. In the present paper, the subvariety lattice of V RC will be characterized.

Irreducible varieties of commutative semigroups

In this paper we describe the varieties of commutative semigroups that are meet- and join-irreducible in the lattice of the varieties of commutative semigroups. We apply the method of A. Kisielewicz [Trans. Amer. Math. Soc. 342 (1994) 275–305]. This leads to investigation of the covering relation in the lattices of remainders and the algebraic structure of the remainders, involving permutation groups acting on the sequences of positive integers. In particular, along the way, we prove a theorem about existence of unique minimal generators for remainders, and provide algorithms to determine all the covers and dual covers of a given variety of commutative semigroups.

Unification in Commutative Semigroups

Unification is one of the basic concepts of automated theorem proving. It concerns such questions as finding solutions of finite sets of equations, determining if every solution comes from a most general solution, and if so, determining how many most general solutions are needed to generate all solutions. These solutions given in terms of substitutions are called, more formally, unifiers. The unification type of a variery (equational class) of algebras is defined according to the cardinality or existence of minimal complete sets of most general unifiers. Of particular interest, from a computational point of view, are varieties of groups and semigroups. So far the problem has been considered mainly for particular varieties. In this paper we determine unification types for all varieties of commutative semigroups. In particular, we prove that for commutative semigroups the unification problem is solvable in the very strong sense that there is an algorithm which for any two finite sets Σ1and Σ2of semigroup equations produces the minimal complete set of the most general unifiers of Σ1over the variety of commutative semigroups generated by Σ2. It seems that this is the first so general decidability result in the area.

Iterative hyperidentities for theA n,m varieties of semigroups

Iterative hyperidentities are hyperidentities of the special formFa(x1,...,xk=Fa+b(x1,...,xk). This type of hyperidentity has been considered by Denecke and Poschel, and by Schweigert. Here we consider iterative hyperidentities for the variety An,m of commutative semigroups satisfyingxn=xn+m,n,m ≥ 1. We introduce two parametersγ(m, n) andβ(m) associated withn andm, and show thatAnn,m satisfies the iterative hyperidentitiesFγ(x1,...,xk =Fγ+b(x1,...,xk) for every arityk. Moreover, the numbersγ andβ are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for An,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.

Embeddability and the word problem

Let be a finitely presented variety with operations Ω and let be the variety having the same set of operations Ω but defined by the empty set of identities. A partial-algebra is a set P with a set of mappings containing for each n-ary operation f of Ω a mapping , where D ⊆ Pn. An incomplete -algebra is a partial -algebra which satisfies the defining identities of , insofar as they can be applied to the partial operations of (Trevor Evans [4, p. 67]). We call an incomplete -algebra a partial Evans-algebra if it can be embedded in a member of the variety .If the class of all partial Evans -algebras is (first-order) finitely axiomatizable, then the word problem for the variety is solvable. (Evans [4, 5]). In 1953 Evans [5, p. 79] raised the question of whether the converse is true. In this paper we show that the answer is in the negative.Let CSg denote the variety of commutative semigroups. We call an incomplete CSg-algebra an incomplete commutative semigroup and we call a partial Evans CSg-algebra a partial Evans commutative semigroup. It is known (A. I. Malcev [9] see also Evans [6]) that the variety of commutative semigroups has solvable word problem. We show (Theorem 1) that the class of all partial Evans commutative semigroups is not finitely axiomatizable. Therefore the solvability of the word problem for the variety of commutative semigroups does not imply finite axiomatizability of the class of all partial Evans commutative semigroups.

Varieties of Commutative Semigroups

Varieties of Commutative Semigroups

On the Joins of Semigroup Varieties with the Variety of Commutative Semigroups

On the Joins of Semigroup Varieties with the Variety of Commutative Semigroups

Varieties of commutative semigroups

In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.

On the joins of semigroup varieties with the variety of commutative semigroups

We show that the join of a variety of semigroups and the variety of all commutative semigroups is not finitely based, provided some weak conditions.

A decision problem for varieties of commutative semigroups

For a first order formula P: ∀x1, …, ∀xn, ∃y1, …, ∃ym (u(x1, …, xn, y1, …, ym) ≡ v(x1, …, xn, y1, …, ym)) where u and v are two words on the alphabet {x1, …, xn, y1, …, ym}, and a finite set E of semigroup identities with xy ≡ yx in it, we prove that it is decidable whether P follows from E, that is whether all the semigroups in the variety defined by E satisfy P.

Hyperidentities for some varieties of commutative semigroups

Hyperidentities and hypervarieties have been defined by Taylor in [5]. A hypervariety is a class of varieties closed under the formation of equivalent, product, reduct and subvarieties. Hyperidentities are used to define hypervarieties, in the same way that ordinary identities define varieties. This paper produces some hyperidentities satisfied by various varieties of commutative semigroups, and identifies some restrictions as to what kind of hyperidentities such varieties can satisfy. It also continues the study, begun in [6], of the closure and hypervariety operators defined there, as they apply to varieties of commutative semigroups.

Some Order Properties of the Lattice of Varieties of Commutative Semigroups

The most complete work on the structure of the lattice of varieties of commutative semigroups available to this date is [12]. Nevertheless, it fails to give the structure of this lattice. In the positive direction, it shows in particular that the order structure of is determined by the order structure of well-known lattices of integers together with the sublattice of varieties of commutative nil semigroups.In the present work, we study from the point of view of order. Perkins [13] has shown that has no infinite descending chains and is countable. The underlying questions we consider here arose from the results of Almeida and Reilly [1] in connection with generalized varieties. There, it is observed that the best-known part of consisting of the varieties all of whose elements are abelian groups is in a sense very wide: it contains infinite subsets of mutually incomparable elements and allows the construction of uncountably many generalized varieties and infinite descending chains of generalized varieties.

Tolerance distributive and tolerance modular varieties of commutative semigroups

Tolerance distributive and tolerance modular varieties of commutative semigroups

The varieties of commutative semigroups for which epis are onto

SynopsisIn a forthcoming paper, N. M. Khan gives a condition for a variety of commutative semigroups V to be saturated in the sense of Howie and Isbell (1967) (i.e. epis are onto for each S ∈ V). We show the necessity of the condition by constructing a non-saturated semigroup which is a member of every commutative variety not satisfying Khan's condition. This determination of the saturated varieties of commutative semigroups enables us then to prove that these varieties form a sublattice of the lattice of varieties of all commutative semigroups.

A note on local polynomial functions on commutative semigroups

AbstractGiven a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

Commutative semigroup varieties with the amalgamation property

AbstractWe determine which varieties of commutative semigroups have the weak or strong amalgamation property. These are precisely the varieties of inflations of semilattices of abelian groups.