ON THE MONOID OF COFINITE PARTIAL ISOMETRIES OF N WITH A BOUNDED FINITE NOISE

Abstract

In the paper we study algebraic properties of the monoid IN ∞ of cofinite partial isometries of the set of positive integers N with the bounded finite noise j. For the monoids IN ∞ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer j every Hausdorff shift-continuous topology τ on IN ∞ is discrete and if IN g[j] ∞ is a proper dense subsemigroup of a Hausdorff semitopological semigroup S, then S \ IN ∞ is a closed ideal of S, and moreover if S is a topological inverse semigroup then S \ IN ∞ is a topological group. Also we describe the algebraic and topological structure of the closure of the monoid IN ∞ in a locally compact topological inverse semigroup.