ON MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY

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