On the decomposable semigroups and applications

TL;DR

This paper introduces the concept of decomposable semigroups to simplify the computation of their ideals, enabling parallel processing and reducing algorithmic complexity using Hermite normal form techniques.

Contribution

It defines decomposable semigroups and varieties, providing a combinatorial characterization and demonstrating how this reduces computational complexity and allows parallel algorithms.

Findings

Decomposable semigroups can be characterized combinatorially.

Computing ideals of decomposable semigroups is simplified and parallelizable.

Applications demonstrate practical benefits of the approach.

Abstract

The aim of this work is to reduce the complexity of the available algorithms for computing the generator sets of a semigroup ideal by using the Hermite normal form. In order to achieve it we introduce the concept of decomposable semigroup. If a semigroup is decomposable, the computation of its ideal is equivalent to compute the ideals of each semigroup in the decomposition, thus obtaining a reduction of the complexity of the algorithms. Furthermore, since these computations are mutually independent, they can be carried out in parallel. The concept of decomposable variety is introduced and a combinatorial characterization of decomposable semigroup is obtained. Some applications are also provided.