Linear representations of inverse semigroups (Q1803363)

Let \(V\) be a linear space over \(\mathbb{C}\). A multi-automorphism of \(V\) is a multi-valued mapping of \(V\) onto itself that respects the linear structure of \(V\). Under natural multiplication, the set \(SML(V)\) of all multi-automorphisms of \(V\) forms an inverse semigroup. The author proves that if \(V_ 1\) and \(V_ 2\) are finite dimensional spaces, then \(SML(V_ 1)\) and \(SML(V_ 2)\) are isomorphic if and only if \(V_ 1\) and \(V_ 2\) are isomorphic. He does not refer to earlier publications of L. B. Shneperman, where stronger results are proved. Another result is a construction of a homomorphic representation of an arbitrary inverse semigroup \(S\) inside \(SML(V)\) for a suitable vector space \(V\). He does not observe that this representation can always be chosen to be faithful.