One-parameter semigroups in a semigroup
Paul S. Mostert, Allen L. Shields · Transactions of the American Mathematical Society · 1960
A topological semigroup is a Hausdorff space S with a continuous, associative multiplication.If there is an identity, 1, then 77(1) will denote the maximal subgroup of S containing 1; in other words, 77(1) is the set of all elements with two-sided inverses.A one-parameter semigroup in 5 is a continuous, one-to-one function a: [0, l]->S such that o-( 0) = 1, and o-(a + b)=o-(a)<r(b) for all a, bE [0, l] for which a+ 6£ [0, l].In this paper we obtain the following result on the existence of oneparameter semigroups. Theorem1. Let S be a compact semigroup with identity, and assume that 77(1) is not an open set in S. Let there be a neighborhood V of the identity containing no other idempotents.Then S contains a one-parameter semigroup a such that <r(a)£77( 1) for 0<a^l.Moreover, a(a)=o-(b)g, gEH(l), implies a = b and g=l.This is a generalization of a previous result of the authors [4, Theorem A] in which it is assumed that 77(1) is a Lie group.Our proof will actually establish the following more general result.Theorem 2. The conclusion of the previous theorem is correct if S is merely assumed to be locally compact, provided 77(1) contains a compact subgroup G open in 77( 1) but not open in S, and provided there is a neighborhood V of G containing no idempotents other than the identity.