arXiv subjects

Gracinda M. S. Gomes

Publications and source records attributed to Gracinda M. S. Gomes.

3 recordsLinked to original sources

arXiv2025

Left reductive regular semigroups

In this paper we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes we observe that the right ideal structure of the semigroup is `embedded' inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of Nambooripad's normal category as our building block, which we call a connected category. The main theorem of the paper describes a category equivalence between the category of left (and right) reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of L- (and R-) unipotent semigroups, right (and left) regular bands, inverse semigroups and arbitrary regular monoids. Finally, we provide concrete (and rather simple) descriptions to the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.

math.GR↗

arXiv2017

On the semigroup rank of a group

For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if $G$ is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if $M$ is a connected closed surface, then $(π_1(M)){\rm rk}S = (π_1(M)){\rm rk}G+1$ if and only if $M$ is orientable.

math.GR↗

arXiv2015

Takahasi semigroups

Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.

math.GR↗