On monoids of injective partial selfmaps almost everywhere the identity

In this paper we study the semigroup (…) of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality (…). We describe the Green relations on (…), all (two-sided) ideals and all congruences of the semigroup (…). We prove that every Hausdorff hereditary Baire topology (…) such that (…) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup (…) in a topological semigroup. Also we show that for an infinite cardinal (…) the discrete semigroup (…) does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning (…) into a topological inverse semigroup.