On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

Abstract

In this paper we study the structure of the monoid IN n ∞ of cofinite partial isometries of the n-th power of the set of positive integers N with the usual metric for a positive integer n 2. We describe the elements of the monoid IN n ∞ as partial transformation of N n , the group of units and the subset of idempotents of the semigroup IN n ∞ , the natural partial order and Green's relations on IN n ∞ . In particular we show that the quotient semigroup IN n ∞ /C mg , where C mg is the minimum group congruence on IN n ∞ , is isomorphic to the symmetric group S n and D = J in IN n ∞ . Also, we prove that for any integer n 2 the semigroup IN n ∞ is isomorphic to the semidirect product S n ⋉ h (P ∞ (N n ), ∪) of the free semilattice with the unit (P ∞ (N n ), ∪) by the symmetric group S n .