Edmond W. H. Lee

SUBVARIETIES OF THE VARIETY GENERATED BY THE FIVE-ELEMENT BRANDT SEMIGROUP

All subvarieties of the variety B2 generated by the five-element Brandt semigroup are characterized. Based on this characterization, an algorithm is provided that decides if a finite set of identities defines, within B2, a finitely generated subvariety.

On the structure of the lattice of combinatorial Rees–Sushkevich varieties

The lattice L(S) of all semigroup varieties is a complex object but some of its parts are rather well understood by now. For instance, the “upper part” of L(S) consists of overcommutative varieties, that is varieties containing the variety of all commutative semigroups. These varieties form a filter in L(S), and it turns out that the filter admits a relatively easy description in terms of congruence lattices of certain unary algebras [39]. In the “lower part” of L(S) consisting of periodic semigroup varieties one can distinguish two important ideals formed by completely regular varieties and by nilsemigroup varieties. The former ideal was investigated in depth by Polak [16, 17, 18] in the 1980s — of course, modulo the lattice of periodic group varieties.

COMBINATORIAL REES-SUSHKEVICH VARIETIES ARE FINITELY BASED

A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. It is shown that all combinatorial Rees–Sushkevich varieties are finitely based.

Limit Varieties Generated by Completely 0-Simple Semigroups

A limit variety is a variety that is minimal with respect to being nonfinitely based. This paper presents a new infinite series of limit semigroup varieties, each of which is generated by a finite 0-simple semigroup with Abelian subgroups. These varieties exhaust all limit varieties generated by completely 0-simple semigroups with Abelian subgroups.

Finitely Generated Limit Varieties of Aperiodic Monoids with Central Idempotents

A non-finitely based variety of algebras is said to be a limit variety if all its proper subvarieties are finitely based. Recently, Marcel Jackson published two examples of finitely generated limit varieties of aperiodic monoids with central idempotents and questioned whether or not they are unique. The present article answers this question affirmatively.

On Identity Bases of Exclusion Varieties for Monoids

An exclusion variety is a variety that is maximal with respect to not containing some semigroup. It is shown that if the bases of all exclusion varieties for some periodic semigroup S are known, then the bases of exclusion varieties for the monoid S 1 can be computed.

Finite basis problem for 2-testable monoids

A monoid S1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.

Small Semigroups Generating Varieties with Continuum Many Subvarieties

The smallest finitely based semigroup currently known to generate a variety with continuum many subvarieties is of order seven. The present article introduces a new example of order six and comments on the possibility of the existence of a smaller example. It is shown that if such an example exists, then up to isomorphism and anti-isomorphism, it must be a unique monoid of order five.

Minimal Semigroups Generating Varieties with Complex Subvariety Lattices

A semigroup is complex if it generates a variety with the property that every finite lattice is embeddable in its subvariety lattice. In this paper, subvariety lattices of varieties generated by small semigroups will be investigated. Specifically, all complex semigroups of minimal order will be identified.

Combinatorial Rees–Sushkevich Varieties That Are Cross, Finitely Generated, Or Small

A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set F of finitely generated varieties constitutes an incomplete sublattice and the set S of small varieties constitutes a strict incomplete sublattice of F . Consequently, a combinatorial Rees– Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set 6 of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity O(nk) where n is the number of identities in 6 and k is the length of the longest word in 6. 2000 Mathematics subject classification: primary 20M07; secondary 03C05, 08B15.