On variants of the extended bicyclic semigroup

Abstract

In the paper we describe the group Aut (C Z ) of automorphisms of the extended bicyclic semigroup C Z and study the variants C m,n Z of the extended bicycle semigroup C Z , where m, n ∈ Z. In particular, we prove that Aut (C Z ) is isomorphic to the additive group of integers, the extended bicyclic semigroup C Z and every its variant are not finitely generated, and describe the subset of idempotents E(C m,n Z ) and Green's relations on the semigroup C m,n Z . Also we show that E(C m,n Z) is an ω-chain and any two variants of the extended bicyclic semigroup C Z are isomorphic. At the end we discuss shiftcontinuous Hausdorff topologies on the variant C 0,0 Z . In particular, we prove that if τ is a Hausdorff shift-continuous topology on C 0,0 Z then each of inequalities a > 0 or b > 0 implies that (a, b) is an isolated point of C 0,0 Z , τ and construct an example of a Hausdorff semigroup topology τ * on the semigroup C 0,0 Z such that all its points with ab 0 and a + b 0 are not isolated in C 0,0 Z , τ * .