Inverse semigroups of homeomorphisms between open subsets
Bridget Bos Baird · Journal of the Australian Mathematical Society · 1977
Abstract The symbolIG(X) denotes the inverse semigroup, under composition of functions, of all homeomorphisms between open subsets of aT1topological spaceX. The first result is that two such semigroupsIG(X) andIG(Y) are isomorphic if and only if the spacesXandYare homeomorphic. Ideals ofIG(X) are next examined and it is shown that for many spacesXthe semigroupIG(X) is 0-simple. We also look at congruences onIG(X); one result is that we determine a congruence which in many instances is the largest proper congruence onIG(X).