Automorphisms on cylindrical semigroups
Joan M. Geramita · Pacific Journal of Mathematics · 1972
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 FλF~ι.The set of all such λ for a given F is called A F .Theorem.Let / and λ be as above.Let G be a compact group.Let 94 J. M. GERAMITA subgroup of S containing an idempotent e is called the group of units of e and denoted H(e).The group of units of 1 is also denoted H(S) and called the group of units of S. If a: S -> S is an automorphism then a(H(S)) = H(S) and a(M(S)) = M(S).NOTATION.The following notation is standard throughout the paper.[α, b]-In a totally ordered set, the closed interval from a to b. ]α, b[-The open interval from a to b. fl-The semigroup of nonnegative real numbers under addition with the usual topology.H*-The one point compactiίication of H, written [0, <>o].H r *-H*j[r, co].Λ-The abstract group of positive real numbers under multiplication.R-The group of real numbers under addition with the usual topology.Z(G)-The center of a group G.[p]-The image of p under the quotient map JEP ->£Γ*.*-As in £*, the closure of BaX, except as noted above for H. X\A-For A c X, the complement of A in X.