The large rank of a finite semigroup using prime subsets

The 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 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.