Semigroups generated by nilpotent transformations (Q580508)

\({\mathcal P}_ X\) is the semigroup of all partial transformations of the set X. The empty transformation is the zero of this semigroup and \({\mathcal L}_ X\) denotes the subsemigroup of \({\mathcal P}_ X\) which is generated by the nilpotent elements. The cardinality of a set A will be denoted by \(| A|\) and the domain and range of a partial transformation \(\alpha\) will be denoted by dom \(\alpha\) and ran \(\alpha\) respectively. The author's goal is to describe the elements in \({\mathcal L}_ X\). He shows that if \(| X|\) is an even integer, then \(\alpha\in {\mathcal L}_ X\) if and only if dom \(\alpha\neq X\). If \(| X|\) is an odd integer n, then \(\alpha\in {\mathcal L}_ X\) if and only if dom \(\alpha\neq X\) and either \(| ran \alpha | \leq n-2\) or \(| ran \alpha | =n-1\) and \(\alpha\) satisfies any one of several additional conditions. For the remainder of the discussion, X will be infinite. We will let \(| X| =k\) and cf(k) will denote the cofinality of k. If \(| ran \alpha | \leq cf(k)\), then \(\alpha\in {\mathcal L}_ X\) if and only if dom \(\alpha\neq X\), \(| X\setminus ran \alpha | \geq cf(k)\) and either \(| X\setminus dom \alpha | \geq cf(k)\) or \(| x\alpha^{-1}| \geq cf(k)\) for some \(x\in X\). If k is singular and \(| ran \alpha | =r\), then \(\alpha \in L_ X\) if and only if dom \(\alpha\neq X\), \(| X\setminus ran \alpha | =k\) and either \(| X\setminus dom \alpha | \geq r\) or for each cardinal number \(s<| ran \alpha |\), \(| x\alpha^{-1}| >s\) for some \(x\in X\). The author then shows that any semigroup can be embedded in a semigroup generated by nilpotents and that an inverse semigroup can be embedded in an inverse semigroup which is generated by nilpotents.