Abstract:
In a way, the semigroup theory started with a result about finite semigroup: the 1928 Suschekewitsch's theorem, describing the structure of finite semigroups without proper ideals. On the 50's, the development of the finite automata theory brought a new motivation for the study of finite semigroups. However, it's only since the 70's that this theory has obtained more and more attention from researchers experiencing a notable growth.
The question of how to calculate the complexity of a finite semigroup S has emerged in the middle 60?s, question that even after forty years, is still the main motivation for many works about finite semigroups. The decidability problem for some pseudovarieties is another problem that still occupies researchers.
In this synthesis work is presented a small study about the ω-word problem in the pseudovarieties Sl, N, K, D, LI, J e LSl. It's also referred a solution of the ω-word problem over R, which has been recently proved as being decidable [13]. The ω-word problem is the problem of deciding, for a pseudovariety and two given ω-terms, if the pseudovariety verifies the equality of the ω-terms.
One of the main interests of this study is related with the fact that the ω-word problem being decidable in V may appear as one of the necessary conditions for the pseudovariety V to be tame. The notion of tameness has recently emerged as a tentative of defining stronger properties others than decidability that could be useful to prove the decidability of pseudovarieties defined by operators, such as the semidirect product, join, Mal'cev product, etc.