The Large Rank of a Finite Semigroup using Prime Subsets

TL;DR

This paper introduces prime subsets as an alternative method to determine the large rank of finite semigroups, providing a shorter proof for Brandt semigroups and calculating the large rank for order-preserving singular selfmaps.

Contribution

It presents a novel approach using prime subsets to find the large rank of finite semigroups, simplifying existing proofs and extending results to new classes.

Findings

Shorter proof of the large rank of Brandt semigroups.

Determination of the large rank of order-preserving singular selfmaps.

Introduction of prime subsets as a useful tool in semigroup theory.

Abstract

The \emph{large rank} of a finite semigroup $\Gamma$, denoted by $r_5(\Gamma)$, is the least number $n$ such that every subset of $\Gamma$ with $n$ elements generates $\Gamma$. Howie and Ribeiro showed that $r_5(\Gamma) = |V| + 1$, where $V$ is a largest proper subsemigroup of $\Gamma$. This work considers the complementary concept of subsemigroups, called \emph{prime subsets}, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro's result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.