Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup. (Q653819)

A semigroup \(S\) of partial transformations on a set \(X\) is called semitransitive if for every \(x,y\in X\) there is \(s\in S\) such that either \(s(x)=y\) or \(s(y)=x\). In the paper under review the authors show that the minimal cardinality of a semitransitive subsemigroup in the singular part of the symmetric inverse semigroup \(\mathcal{IS}_n\) equals \(2n-p+1\), where \(p\) is the greatest proper divisor of \(n\). Furthermore, the authors provide a complete classification and construction of all semitransitive subsemigroups of this minimal cardinality \(2n-p+1\).