Hyperassociative varieties of semigroups

A variety of semigroups is calledhyperassociative if the associative law is satisfied as a hyperidentity. We give an equational basis for the variety of all hyperassociative semigroups.     

Solid varieties of arbitrary type

A solid variety is an equational class in which every identity holds as a hyperidentity as well, meaning that it is satisfied not just by the fundamental operations but also by all terms of the appropriate arity. For type (2), an infinite number of solid varieties (of semigroups) are known, but for other types very few examples of solid varieties are known. In this paper we present several constructions which produce infinite chains of solid varieties. One construction generalizes the normalization of a variety, and gives a method to produce a chain of solid varieties from any given solid variety of type (n). The second construction generalizes the rectangular nilpotent varieties of type (2) to type (n). Finally, we use identities which are consequences of idempotency to construct an infinite chain of solid varieties of any fixed type. Received March 23, 2001; accepted in final form July 11, 2002. RID="h1" ID="h1"Research of the third author was supported by NSERC of Canada.     

The Order of Hypersubstitutions of Type （2, 2）

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type （2, 2）, i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type （2, 2）. The main result is that the order is 1, 2, 3, 4 or infinite.     

Varieties generated by unstable involution semigroups with continuum many subvarieties

Over the years, several finite semigroups have been found to generate varieties with continuum many subvarieties. However, finite involution semigroups that generate varieties with continuum many subvarieties seem much rarer; in fact, only one example—an inverse semigroup of order 165—has so far been published. Nevertheless, it is shown in the present article that there are many smaller examples among involution semigroups that are unstable in the sense that the varieties they generate contain some involution semilattice with nontrivial unary operation. The most prominent examples are the unstable finite involution semigroups that are inherently non-finitely based, the smallest ones of which are of order six. It follows that the join of two finitely generated varieties of involution semigroups with finitely many subvarieties can contain continuum many subvarieties.     

Hyperidentities and solid varieties

Let V be a variety of type τ. A type τ hyperidentity of V is an identity of V which also holds in an additional stronger sense: for every substitution of terms of the variety (of appropriate arity) for the operation symbols in the identity, the resulting equation holds as an identity of the variety. Such identities were first introduced by Walter Taylor in [27] in 1981. A variety is called solid if all its identities also hold as hyperidentities. For example, the semigroup variety of rectangular bands is a solid variety. For any fixed type τ, the collection of all solid varieties of type τ forms a complete lattice which is a sublattice of the lattice L(τ) of all varieties of type τ. In this paper we give an overview of the study of hyperidentities and solid varieties, particularly for varieties of semigroups, culminating in the construction of an infinite collection of solid varieties of arbitrary type. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 3, 2006. This paper is an expanded version of a talk presented at the Conference on Algebras, Lattices and Varieties in Honour of Walter Taylor, in Boulder Colorado, August 2004. The author’s research is supported by NSERC of Canada.     

On finite hyperidentity bases for varieties of semigroups

For any varietyV of semigroups, we denote byH(V) the collection of all hyperidentities satisfied byV. It is natural to ask whether, for a givenV, 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 varietyS 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 ofS, namely all semigroup varieties except those satisfyingx ² =x ^2+m for somem. As a first step in the investigation of these exceptional varieties, we also present some hyperidentities satisfied by the variety B_1,1 of bands, and, using the same technique, show thatH(V) is not finitely based for any subvarietyV of B_1,1. 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.Presented by W. Taylor.Research supported by Natural Sciences & Engineering Research Council of Canada.     

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.Presented by Walter Taylor.The results described in this paper form part of the author's Ph.D thesis, submitted to Simon Fraser University, Burnaby, Canada. The author is grateful for the help of her supervisor, Dr. N. R. Reilly, and for the financial support received from Simon Fraser University.     

Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This ‘growth alternative’ conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied.     

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

We use classical results on the lattice of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps (DA) of subpseudovarieties of DA, – where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps (DA), whose quotient is isomorphic to , and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V _k be the pseudovariety generated by the k-generated elements of DA (k≥ 1), we use all our results to compute the position of the congruence class of V _k in . Received April 24, 1996; accepted in final form April 3, 1997.     

Varieties and radicals of near-rings

It is shown that a variety of associative rings which has attainable identities in the class of all associatives ring has attainable identities in the class of all near—rings. We also give examples of varieties of near-rings which are, contrary to the ring case, closed under extensions but do not have attainable identities and varieties which are closed under extensions but not closed under essential extensions, respectively.     

Normal forms for semigroup amalgams

We define a class of inverse semigroup amalgams and derive normal forms for the amalgamated free products in the variety of semigroups. The class includes all amalgams of finite inverse semigroups, recently studied by Cherubini, Jajcayova, Meakin, Nuccio, Piochi and Rodaro (2005–2014), and lower bounded amalgams, that were introduced by the author (1997). We provide sufficient conditions for decidable word problem. We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups. The normal forms can be used to study amalgams in subvarieties of inverse semigroups. In a forthcoming paper by the author, the results are used for varieties of semilattices of groups.     

Varieties and Nilpotent Extensions of Permutative Periodic Semigroups

We investigate certain semigroup varieties formed by nilpotent extensions of orthodox normal bands of commutative periodic groups. Such semigroups are shown to be both structurally periodic and structurally commutative, and are therefore structurally inverse semigroups. Such semigroups are also shown to be dense semilattices of structurally group semigroups. Making use of these structure decompositions, we prove that the subvariety lattice of any variety comprised of such semigroups is isomorphic to the direct product of the following three sublattices: its sublattice of all structurally trivial semigroup varieties, its sublattice of all semilattice varieties, and its sublattice of all group varieties. We conclude, therefore, that to completely describe this lattice, we must first describe completely the lattice of all structurally trivial semigroup varieties, since the other two are well known lattices.     

Regular principal factors in free objects in Rees-Sushkevich varieties

In a previous paper, the author showed how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges. In the main theorem, it is shown how these fundamental semigroups can be used to describe the regular principal factors of the free objects in certain Rees-Sushkevich varieties, namely, the varieties of semigroups that are generated by all completely 0-simple semigroups over groups in a variety of finite exponent. This approach is then used to solve the word problem for each of these varieties for which the corresponding group variety has solvable word problem.     

Null Extensions and Generalized Inflations of Brandt Semigroups

Clarke and Monzo defined in [3] a construction called a generalized inflation of a semigroup. It is always the case that any inflation of a semigroup is a generalized inflation, and any generalized inflation of a semigroup is a null extension of the semigroup. Clarke and Monzo proved that any associative null extension of a base semigroup which is a union of groups is in fact a generalized inflation. In this paper we study null extensions and generalized inflations of Brandt semigroups. We first prove that any generalized inflation of a Brandt semigroup is actually an inflation of the semigroup. This answers a question posed by Clarke and Monzo in [3]. Then we characterize associative null extensions of Brandt semigroups, and show that there are associative null extensions of Brandt semigroups which are not generalized inflations.     

Proper two-sided restriction semigroups and partial actions

Two-sided restriction semigroups and their handed versions arise from a number of sources. Attracting a deal of recent interest, they appear under a plethora of names in the literature. The class of left restriction semigroups essentially provides an axiomatisation of semigroups of partial mappings. It is known that this class admits proper covers, and that proper left restriction semigroups can be described by monoids acting on the left of semilattices. Any proper left restriction semigroup embeds into a semidirect product of a semilattice by a monoid, and moreover, this result is known in the wider context of left restriction categories. The dual results hold for right restriction semigroups.What can we say about two-sided restriction semigroups, hereafter referred to simply as restriction semigroups? Certainly, proper covers are known to exist. Here we consider whether proper restriction semigroups can be described in a natural way by monoids acting on both sides of a semilattice.It transpires that to obtain the full class of proper restriction semigroups, we must use partial actions of monoids, thus recovering results of Petrich and Reilly and of Lawson for inverse semigroups and ample semigroups, respectively. We also describe the class of proper restriction semigroups such that the partial actions can be mutually extendable to actions. Proper inverse and free restriction semigroups (which are proper) have this form, but we give examples of proper restriction semigroups which do not.     

Cancellation law and attainable classes of linear ω-algebras

With the aid of mixed linear Ω-algebras we prove a theorem to the effect that the cancellation law is satisfied in a groupoid of subvarieties of a variety of Ω-algebras linear over a field and given by identities of zero order. We show that in some varieties of Ω-algebras linear over an infinite ring of principal ideals there are no nontrivial finitely attainable subvarieties. As examples of such varieties we cite the varieties of all Ω-rings, of all rings, of commutative or anticommutative rings (Ω-rings), of Lie rings, et al. In the case of anticommutative rings (Ω-rings) this property holds for Ω-algebras, linear over an arbitrary integral domain without stable ideals.