The Large Rank of a Finite Semigroup using Prime Subsets

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.

Semigroup Forum, To appear

References in corpus (2)

• The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup
• Rank Properties of Multiplicative Semigroup Reduct of Affine Near-Semirings over $B_n$

Cited by in corpus (1)

• Rank Properties of Multiplicative Semigroup Reduct of Affine Near-Semirings over $B_n$