Shuhua Zhang

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

Complete endomorphisms of the lattice of pseudovarieties of finite semigroups

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

Operators on the Lattice of Pseudovarieties of Finite Semigroups

for some special class

On Radical Congruence Systems II

{\cal A} of finite semigroups. In some instances the class is itself a pseudovariety while in others the class will be defined in terms of certain congruences associated with Greens relations. The basic properties of these complete congruences are presented and some relations between certain operators associated with these congruences are obtained.

Commutativity of operators on the lattice of existence varieties

In a series of recent papers, certain techniques that have proved valuable in the study of the lattice of varieties of completely regular semigroups (see Pastijn [12], Polak [20], Petrich and Reilly [16, 17]) involving the existence of certain complete congruences have been extended, first to the lattice of e-varieties of regular semigroups (Reilly and Zhang [24]) and then to the study of the lattice £(F) of pseudovarieties of finite semigroups (Auinger, Hall, Reilly and Zhang [4]). In particular, it is shown in [4] that there exists a family of complete congruences of the form:

Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [Uρ, Uρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of Uρ, bases of pseudoidentities for Uρ, instances of commutativity (Uρ)σ = Uσ)ρ, as well as the semigroups generated by certain pairs of such operators.

Congruence Relations on the Lattice of Existence Varieties of Regular Semigroups

This paper is a continuation of a paper of the same title by the first author and P. Weil. We first characterize the universal class of a radical congruence system. We then introduce the meet and the (limit) iteration of congruence systems. This enables us to generate new radical congruence systems from given congruence systems. Some interesting examples are presented. We finally determine the smallest radical congruence systems whose universal classes are N, LZ ◦ N, RZ ◦ N, and RB ◦ N respectively.

Completely regular semigroup varieties generated by mal'cev products with groups

In a previous paper the authors introduced seven complete congruences on the lattice ℒev(ℛI of e-varieties of regular semigroups of the form ρP:UρPV⇔P∘U=P∘V, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form αP:UαPV⇔P∩U=P∩V, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence ρ on ℒev(ℛI) and eachU∈ℒev(ℛI), the ρ-class ofU is an interval [Uρ,Uρ] so that there is associated with each such congruence an idempotent operatorU→Uρ on ℒev(ℛI). This paper establishes numerous results concerning the commutativity of operators of this form.