A nonstandard approach to compact inverse semigroups of local homeomorphisms (Q1375887)

At the beginning, a brief survey of a nonstandard approach to convergence structure on an algebra is given. The latter half is mainly concerned with compact inverse semigroups. This part includes some useful facts of information. The basic concept is a homomorphism \(P\) of a convergence inverse semigroup \(S= (S,\cdot,{}^{-1}, \sigma)\) to the symmetric convergence inverse semigroup which is called a representation of convergence inverse semigroup \(S\) by local homeomorphisms of a convergence space \((A,\alpha)\). A symmetric convergence inverse semigroup (\(LH(A)\), a composition \(\circ\), inverse of relation \(-1\), continuous convergence \(\gamma\)) is formed by the set of all local homeomorphisms of \(A\) to itself. This is defined by: if any \(g\in S\), \(x\in\) the nonstandard extension \({}^*S\) of \(S\), (\(g,x)\in\sigma \Leftrightarrow (P(g),P(x))\in \gamma\). If the homomorphism is one-to-one, the representation is called faithful. Let us mention some of the obtained results. If \(S\) has only a finite number of idempotents, then \(S\) has a faithful representation by local homeomorphisms of a compact \(T_2\) convergence space. This part of the conclusion is equivalent to: \(S\) satisfies the condition: \((s,x)\in \sigma\) and \((hs,z)\in\sigma \Rightarrow (\exists t\in {}^*S)z= tx\). The main result is read as: any compact \(T_2\) convergence inverse semigroup is homeomorphic to a convergence inverse semigroup of local homeomorphisms of a convergence space.