Vol. 43, No. 1, 1972

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Automorphisms on cylindrical semigroups

Joan Geramita

Vol. 43 (1972), No. 1, 93–105

This paper characterizes the automorphisms of a cylindrical semigroup S in terms of the automorphisms of the defining subgroups and subsemigroups. The following theorem is representative of the type of information given in this paper.

Let F : R A be a dense homomorphism of the additive real numbers to the compact abelian group A. Let λ be a positive real number. Multiplication by λ shall also denote the automorphism of A whose restriction to F(R) is given by FλF1. The set of all such λ for a given F is called Λ1Γ.

Theorem. Let f and λ be as above. Let G be a compact group. Let

S = {(p,f(p)g) : p ∈ H and g ∈ G }∪ α× A × G.

Then α;S S is an automorphism if and only if α(p,f(p),g) = (λp,f(λp)(f(p))ξ(g));α(,a,g) = (,λa,τ(a)ξ(g)), where τ : A G is a homomorphism into the centre of G and, ξ : G G is an automorphism. Theorem. Let S be as in theorem above. Let 𝒜(G) be the automorphism group of G, and Z(G), the center of G. The automorphism group of S is isomorphic as an abstract group to 𝒜(G) × (AF × Hom(A,Z(G))) with the following multiplication

        -  --        -   --   --
(ξ,(λ,τ))(ξ,(λ,τ)) = (ξ ∘ξ,(λλ,(τ ∘λ )(ξ ∘ τ))).

Mathematical Subject Classification 2000
Primary: 22A15
Received: 15 July 1971
Revised: 28 September 1971
Published: 1 October 1972
Joan Geramita