Endomorphisms of finite symmetric inverse semigroups
Denote by \({\mathcal I}_n\) the symmetric inverse semigroup on \(X=\{1,2,\dots,n\}\). For any function \(f\), the rank of \(f\) is denoted by \(\text{rank}(f)\) and is the cardinality of the range of \(f\). For \(1\leq k\leq n+1\), let \(I_k=\{t\in{\mathcal I}_n:\text{rank}(t)<k\}\). Finally denote by \({\mathcal G}_n\) the group of units of \({\mathcal I}_n\), that is, the group of all permutations of \(X\). It has long been known that all the automorphisms of \({\mathcal I}_n\) are inner but until this paper, nothing appears to have been known about the endomorphisms of \({\mathcal I}_n\). They are completely described here and the main result and its consequences are quite interesting. For example, suppose \(n>2\) and \(n\neq 4\). If \(\varphi\) is an endomorphism of \({\mathcal I}_n\) which is not an automorphism and \(\varphi[{\mathcal I}_n]\) contains more than three elements, then either \(\varphi[{\mathcal I}_n]={\mathcal G}_n\cup\{\emptyset\}\) or \(\varphi[{\mathcal I}_n]={\mathcal G}_n\cup I_2\). It turns out that \({\mathcal I}_4\) has additional endomorphisms which are different in nature from the endomorphisms of \({\mathcal I}_n\) when \(n>2\) and \(n\neq 4\). The authors also determine the ranks of the endomorphisms. If \(n>2\) and \(n\neq 4\), then the rank of any endomorphism is either 1, 2, 3, \(n!+1\), or \(n!+n^2+1\). \({\mathcal I}_4\) has, in addition to these, endomorphisms of rank 7. Finally, they determine the number of endomorphisms of \({\mathcal I}_n\) and they characterize those which are idempotent.
scientific article; zbMATH DE number 1106642