Maximal subsemigroups of finite transformation semigroups \(K(n,r)\). (Q1882492)

In this paper we find a description of all maximal subsemigroups of \(K(n,r)\), the subsemigroup of the full transformation semigroup \(T_n\), consisting of mappings of rank no more than \(r\), where \(2\leq r\leq n-1\). These fall into three types, the first two of which are regular while the third is non-regular but abundant. The total number of maximal subsemigroups is found as a certain sum in \(n\) and \(r\).