The Large Rank of a Finite Semigroup using Prime Subsets

The \emph{large rank} of a finite semigroup $Γ$, denoted by $r_5(Γ)$, is the least number $n$ such that every subset of $Γ$ with $n$ elements generates $Γ$. Howie and Ribeiro showed that $r_5(Γ) = |V| + 1$, where $V$ is a largest proper subsemigroup of $Γ$. 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.

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