The endomorphisms of a finite chain form a Rees congruence semigroup
Let \(O_n\) denote the semigroup of all order-preserving mappings on the finite chain \(1<2<\cdots<n\), under composition, that is to say the semigroup of all endomorphisms of the finite chain. It is proved that every non-identity congruence on \(O_n\) is a Rees congruence, meaning that the only non-singleton congruence class is an ideal. For this semigroup of transformations, the ideals themselves form a chain \(I_1\subseteq I_2\subseteq\cdots\subseteq I_n\), where \(I_k\) consists of all mappings of rank \(k\).
scientific article; zbMATH DE number 1330704