In this paper; we study equations on semidirect products of commutative semigroups. Let Comq,r denote the pseudovariety of all finite semigroups that satisfy the equations xy = yx and x r + q = xr. The pseudovariety Com1,1 is the pseudovariety of all finite semilattices. We consider the product pseudovariety Comq,r * generated by all semidirect products of the form S*T with S ∈ Comq,r and T ∈ ,...

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not fi...

Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial pro...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the pseudovariety H of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon’s approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix...

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute J-pointlike sets, where J denotes the pseudovariety o...

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set π of primes; the pseudovariety of all finite semigroups in which every regular J-class is the product of a rectangular band by a group from...

A new topological representation for free profinite R-trivial semigroups in terms of spaces of vertex-labeled complete binary trees is obtained. Such a tree may be naturally folded into a finite automaton if and only if the element it represents is an ω-term. The variety of ω-semigroups generated by all finite R-trivial semigroups, with the usual interpretation of the ω-power, is then studied. ...