Local finiteness for Green relations in (I-)semigroup varieties

TL;DR

This paper investigates the property of local finiteness related to Green's relations within the lattice of semigroup and I-semigroup varieties, focusing on how finitely generated semigroups exhibit finite K-classes.

Contribution

It introduces the concept of local K-finiteness for varieties of (I-)semigroups and analyzes its implications within the lattice structure of these varieties.

Findings

Characterization of locally K-finite varieties

Identification of conditions for finiteness of K-classes

Insights into the structure of semigroup varieties

Abstract

In this work, the lattice of varieties of semigroups and the lattice of varieties of I-semigroups (a common setting for both the variety of completely regular semigroups and the variety of inverse semigroups) are studied with respect to the following concepts: a variety V of (I-)semigroups is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many (distinct) K-classes.