Operators on the Lattice of Pseudovarieties of Finite Semigroups

Abstract

As the first 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, 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 [U ρ , U ρ ]. This provides a family of operators of the form U → U ρ on L(F ) that reveal important relationships between elements of L(F ). Various aspects of these operators are considered including characterizations of U ρ , bases of pseudoidentities for U ρ , instances of commutativity (U ρ ) σ = (U σ ) ρ , as well as the semigroups generated by certain pairs of such operators.

Complete endomorphisms of the lattice of pseudovarieties of finite semigroups

“…Then S is called group closed if G r (S), the set of all group elements of S, is a subsemigroup (which then is clearly completely regular) of S. Denote the class of all finite group closed semigroups by GC. By [25,Lemma 3.2], GC = [x u+1 y u>+1 = (x u+1 y u+1 )" +1 ] so that GC is a pseudovariety. The reverse containment holds trivially.…”

“…where o denotes the Mal'cev product and P is any one of the eight pseudovarieties interesting interplay between the lattice theoretic structure of £(F) and the Mal'cev product and leads naturally to the study of operators on £(F) of the form V -> P o V. This study was pursued by the present authors in [25]. Operators on £(F) of the form V -> V f l W, for certain special pseudovarieties W , such as the pseudovariety of groups, bands, et cetera also play an important role in the study of £(F) and are the main focus of this paper.…”

Idempotent-generated semigroups and pseudovarieties

“…The more general question can be viewed as one of many similar questions arising from [4]. See [16] for related questions.…”

On Radical Congruence Systems II

“…In the following list we give some important pseudovarieties which will be used in the sequel. For the description by pseudoidentities, the interested reader is referred to Almeida [1], Pin [17] and Reilly and Zhang [20]:…”

“…The following basic, but important facts concerning some subpseudovarieties of LG will be useful in the sequel. The interested reader is referred to Almeida [1], Hall and Weil [11], Pin [17] and Reilly and Zhang [20] for proofs. It should be pointed out that for any nontrivial group pseudovariety H, (RB V H) o N is always a proper subpseudovariety of LH (see [11,Corollary 3.2]).…”