Rank Properties of the Semigroup of Endomorphisms over Brandt semigroup

TL;DR

This paper investigates various rank properties of the semigroup of endomorphisms over Brandt semigroups, extending the understanding of their algebraic structure.

Contribution

It provides a comprehensive analysis of all rank types for the endomorphism semigroup over Brandt semigroups, a topic not previously explored in detail.

Findings

Determined the small, lower, intermediate, upper, and large ranks of the semigroup.

Established relationships between different rank types for these semigroups.

Extended the theory of ranks to a new class of semigroups.

Abstract

Howie and Ribeiro \cite{a.Howie99,a.Howie00} introduced various ranks, viz. small rank, lower rank, intermediate rank, upper rank and the large rank of a finite semigroup. In this note, we investigate all these ranks of the semigroup of endomorphisms over Brandt semigroup.