The joint of the pseudovarieties of idempotent semigroups and locally trivial semigroups

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.

Operators on the Lattice of Pseudovarieties of Finite Semigroups

Reilly¹,

Zhang²

1998

Three Examples of Join Computations

Azevedo¹,

Zeitoun²

1998

Some pseudovariety joins involvinglocally trivial semigroups

Costa¹

2001

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety LI of locally trivial semigroups. We compute, in particular, the join of LI with any subpseudovariety of CR h m N, the Mal'cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS = e (resp. Se = e) for each idempotent e of S.