The $\mathcal{R}$-height of semigroups and their bi-ideals

TL;DR

This paper investigates the $ $-height of semigroups and their ideals, establishing bounds and exploring whether these bounds are attainable, thereby advancing understanding of the structure of semigroups.

Contribution

The paper provides bounds on the $ $-height of bi-ideals and other ideals in semigroups with finite $ $-height, and examines the attainability of these bounds.

Findings

Bi-ideals inherit finite $ $-height from the semigroup.

Bounds on $ $-height of various ideals are established.

Questions about the attainability of these bounds are explored.

Abstract

The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we establish bounds on the $\mathcal{R}$-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite $\mathcal{R}$-height. We then investigate whether these bounds can be attained.