Complete Congruences on the Lattice of Rees–Sushkevich Varieties

We study the lattice ?(RS_n) of subvarieties of the variety of semigroups generated by completely 0-simple semigroups over groups with exponent dividing n, with a particular focus on the lattice ??(RS_n) consisting of those varieties that are generated by completely 0-simple semigroups. The sublattice of ??(RS_n) consisting of the aperiodic varieties is described and several endomorphisms of ?(RS_n) considered. The complete congruence on ??(RS_n) that relates varieties containing the same aperiodic completely 0-simple semigroups is considered in some detail.     

Identity Bases for Some Non-exact Varieties

Let A₂ be the variety generated by the five-element non-orthodox 0-simple semigroup. This paper presents the identity bases for several subvarieties of A₂ that are not generated by any completely 0-simple or completely simple semigroups. It will be shown that several subvarieties of A₂, including the variety generated by the five-element Brandt semigroup, are hereditarily finitely based.     

Shades of orthodoxy in Rees-Sushkevich varieties

It is shown that, within the class of Rees-Sushkevich varieties that are generated by completely (0-) simple semigroups over groups of exponent dividing n, there is a hierarchy of varieties determined by the lengths of the products of idempotents that will, if they fall into a group ℋ-class, be idempotent. Moreover, the lattice of varieties generated by completely (0-) simple semigroups over groups of exponent dividing n, with the property that all products of idempotents that fall into group ℋ-classes are idempotent, is shown to be isomorphic to the direct product of the lattice of varieties of groups with exponent dividing n and the lattice of exact subvarieties of a variety generated by a certain five element completely 0-simple semigroup.     

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.     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

A JUST NON-FINITELY BASED VARIETY OF BIGROUPS

A bigroup is a pair (H, π) consisting of a group H and an idempotent endomorphism π of H. One can consider π as a unary operation on H so a bigroup is a universal algebra. The aim of our paper is to construct the first example of a just non-finitely based variety of bigroups i.e. a variety which is non-finitely based but all whose proper subvarieties are finitely based. There is a close similarity between varieties of bigroups and varieties of groups so we hope that our result could help to construct a just non-finitely based variety of groups.

     

Identities of semigroup rings over semilattices of completely (0-) simple semigroups

LetR be a ring with identity,S be a semigroup with the set of idempotentsE(S), and denote (E(S)) for the subsemigroup ofS generated byE(S). In this paper, we prove that ifS is a semilattice of completely 0-simple semigroups and completely simple semigroups, then the semigroup ringRS possesses an identity iff so doesR(E(S)); especially, the result is true forS being a completely regular semigroup.     

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.     

Discriminator or dual discriminator?

Varieties are considered with p(x, y, z), a single ternary operation, which acts as a local discriminator or dual discriminator on the subdirectly irreducible elements. If p(x, y, z) is "global", then all subvarieties are finitely based. In the general case a continuum of non-finitely based subvarieties are presented. A graph theoretical picture leads to a variety of groupoids connecting the left-zero and the right-zero semigroups. For this variety some open problems are presented. Received October 7, 1998; accepted in final form October 4, 1999.     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

Chain decompositions of ordered semigroups

Characterizations of ordered semigroups which can be decomposed into (natural ordered) chains of ω -simple ordered semigroups are given, where ω -simple ordered semigroups are ξ _l (ξ _t ) -simple, left (t -) simple, L _n (H _n ) -simple, l (t )-archimedean and nil-extensions of left (t -) simple ordered semigroups, respectively. As a generalization of the theory of Clifford semigroups (without orders) to ordered semigroups, ordered semigroups which are semilattices of t -simple subsemigroups are characterized.     

Finitely based finite involution semigroups with non-finitely based reducts

Since the study of the finite basis problem for finite semigroups began in the 1960s, it has been unknown if there exists any finite involution semigroup that is finitely based but the reduct of which is non-finitely based. The present article exhibits an example of such an involution semigroup of order n+5 for any positive integer n.     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

Generating Sets of Completely 0-Simple Semigroups

ABSTRACT

A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ?⁰[G; I, Λ; P], is given. The result generalizes that of Ru?kuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I|, |Λ|, the number of connected components k of P, and a number r _min, which we define. We go on to show that the number r _min is expressible in terms of a family of subgroups of G, the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P. A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G,{1,…,n}).     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

On the Finite and Nonfinite Generation of Diagonal Acts

The diagonal right act of a semigroup S is the set S × S, with S acting by componentwise multiplication from the right. The diagonal left act and diagonal bi-act of S are defined analogously.

Necessary and sufficient conditions are found for the finite generation of the diagonal bi-acts of completely zero-simple semigroups and completely simple semigroups. It is also proved that various classes of semigroups do not have finitely generated or cyclic diagonal right, left, or bi-acts.     

Bipartite graphs and completely 0-simple semigroups

In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases.