Endomorphisms of the semigroup of order-preserving mappings.
An endomorphism \(f\) of \(\{1,2,\dots,n\}\) is called order preserving provided that \(i\leq j\) implies \(f(i)\leq f(j)\). The main result of this paper gives a complete classification of the semigroup of all order-preserving endomorphisms of \(\{1,2,\dots,n\}\). It turns out that every endomorphism \(\varphi\) of this semigroup has one of the following three types: (i) \(\varphi\) is an automorphism (and hence is either the identity or the conjugation by the order reversing permutation); (ii) \(\varphi\) is constant (and hence maps the whole semigroup to an idempotent); (iii) the image of \(\varphi\) contains exactly two elements, both these elements are idempotents, one of them is the image of the identity (which coincides with the group of invertible elements) and the other one is the image of the singular part.
scientific article; zbMATH DE number 5806755