A classification of the maximal idempotent-generated subsemigroups of finite singular semigroups
Let \(T_n\) be the full transformation semigroup on a set \(X_n\) of \(n\) letters, i.e., the set of all maps \(\alpha\colon X_n\to X_n\) under the operation of composition. For \(1\leq r\leq n\), let \(K(n,r)=\{\alpha\in T_n:|\text{im}(\alpha)|\leq r\}\). Each \(K(n,r)\) is a two-sided ideal of \(T_n\), and \(K(n,n-1)\) is the singular subsemigroup of \(T_n\). Various properties of \(K(n,n-1)\) are known. Among other things, it is well known that the singular subsemigroup is idempotent-generated. The authors calculate the number of and completely classify the maximal idempotent-generated subsemigroups of \(K(n,n-1)\).
scientific article; zbMATH DE number 1747979