Norman R. Reilly

Completely Regular Semigroups

A semigroup S is completely regular if and only if it is a disjoint union of groups. This concept, so simple in its formulation, has intrigued investigators for over forty years. In the early days, the underlying proposition was that the theory of such objects must necessarily be readily derivable from that for groups. The early work of Rees and Clifford gave some support to this notion. However, the work of recent years, especially that on varieties, has shown that the study of completely regular semigroups requires its own ingenious arsenal of tools.

Free generators in free inverse semigroups

Using the characterization of the free inverse semigroup F on a set X , given by Scheiblich, a necessary and sufficient condition is found for a subset K of an inverse semigroup S to be a set of free generators for the inverse sub semigroup of S generated by K . It is then shown that any non-idempotent element of F generates the free inverse semigroup on one generator and that if | X | > 2 then F contains the free inverse semigroup on a countable number of generators. In addition, it is shown that if | X | = 1 then F does not contain the free inverse semigroup on two generators.

Varieties of completely regular semigroups

If CS (respectively, O ) denotes the class of all completely simple semigroups (respectively, semigroups that are orthodox unions of groups) then CS (respectively, O ) is a variety of algebras with respect to the operations of multiplication and inversion. The main result shows that the lattice of subvarieties of is a precisely determined subdirect product of the lattice of subvarieties of CS and the lattice of subvarieties of O . A basis of identities is obtained for any variety in terms of bases of identities for . Several operators on the lattice of subvarieties of are also introduced and studied.

E -unitary congruences on inverse semigroups

By an E -unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest. An important structure theorem for E -unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G , a partially ordered set and a subset , satisfying certain conditions, he constructs an E -unitary inverse semigroup P ( G , ). A semigroup of this type is called a P -semigroup. The structure theorem states that every E -unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlisters Theorems A and B respectively. A triple ( C , ) of the type used to construct a P -semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E -unitary inverse semigroup.

Varieties of lattice-ordered groups in which prime powers commute

We show that the set of all varieties of lattice ordered groups contained in the metabelian variety defined by the lawxpyp=ypxp (wherep is a prime) is a well ordered tower. We give an explicit construction of the subdirectly irreducible members of each of these varieties and show that each variety is defined by a single equation.

Varieties of groups and of completely simple semigroups

Completely simple semigroups form a variety if we consider them both with the multiplication and the operation of inversion. Denote the lattice of all varieties of completely simple semi-groups by L ( CS ) and that of varieties of groups by L ( G ). We prove that the mappings V → V ∩ G and V → V v G are homomorphisms of L ( CS ) onto L ( G ) and the interval [ G, CS ], respectively. The homomorphism V → ( V ∩ G, V v G ) is an isomorphism of L ( CS ) onto a subdirect product. We explore different properties of the congruences on L ( CS ) induced by these homomorphisms.

Operators related to idempotent generated and monoid completely regular semigroups

The class CR of completely regular semigroups (unions of groups or algebras with the associative binary operation of multiplication and the unary operation of inversion subject to the laws x = xx -1 , (x −1 ) -1 = x and xx -1 = x -1 x) is a variety. Among the important subclasses of CR are the classes M of monoids and I of idempotent generated members. For each C ∈ { I, M }, there are associated mappings ν → ν ∩ C and ν → (Ν ∩ C ), the variety generated by ν ∩ C . The lattice theoretic properties of these mappings and the interactions between these mappings are studied.

Congruences on the Lattice of Pseudovarieties of Finite Semigroups

As a step in a study of the lattice ℒ(ℱ) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, a family of complete congruences is introduced on ℒ(ℱ). Such congruences provide a framework from which to study ℒ(ℱ) both locally and globally. Each is associated with a mapping of the form

Varieties of lattice ordered groups that contain no non-abelian o-groups are solvable

{\cal U} \to {\cal U} \cap {\cal A}

Bisimple inverse semigroups as semigroups of ordered triples

for some special class