Decidability and Tameness in the Theory of Finite Semigroups

Decidability and Tameness in the Theory of Finite Semigroups

Tameness of The Pseudovariety Ls1

A class of finite semigroups V is said to be decidable if the membership problem for V has a solution, that is, if we can construct an algorithm to test whether a given semigroup lies in V. Decidability of pseudovarieties is not preserved by some of the most common pseudovariety operators, such as semidirect product, Mal’cev product and join [1, 17]. In particular Rhodes [17] has exhibited a de...

Tameness of pseudovariety joins

The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg [1] as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties. Recall that the join V ∨ W of two pseudovarieties V and W is the least pseudovariety containing both V and W. This talk is concerned with the problem of proving tameness of joins. This problem was consider...

An automata-theoretic approach to the word problem for ω-terms over R

The main contribution of this paper is the solution of a word problem over R, the pseudovariety of all R-trivial semigroups which corresponds, in Eilenberg’s correspondence, to disjoint unions of languages of the form A0a1A ∗ 1a2 . . . anA ∗ n, where the ai’s are letters and ai / ∈ Ai−1 for 1 6 i 6 n. They are also the divisors of transition semigroups of the so-called extensive automata, that ...

Globals of Pseudovarieties of Commutative Semigroups: the Finite Basis Problem, Decidability, and Gaps

Whereas pseudovarieties of commutative semigroups are known to be nitely based, the globals of monoidal pseudovarieties of commutative semi-groups are shown to be nitely based (or of nite vertex rank) if and only if the index is 0, 1 or !. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are sh...