A semigroup associated with a transformation group

Abstract

Let (X, T, wx) be a transformation group with compact Hausdorff phase space X, and let G = [7rt/t TI be the transition group of (X, T, ir). Then G is a group of homeomorphisms of X onto X and so may be regarded as a subset of Xx. The enveloping semigroup E of (X, T, r) is by definition the closure of G in XX [2]. In the first half of this paper algebraic properties of E are studied and correlated with recursive properties of T. Here the main theorem states that proximal is an equivalence relation in X if and only if there is only one minimal right ideal in E. The latter half of the paper is concerned with the study of homomorphic images of transformation groups by means of their enveloping semigroups. For further reference see [2] and [3]. Topological semigroups occur in the literature; see [4]. However, the assumption is generally made that the semigroup multiplication is bilaterally continuous. This is a property which the multiplication in E does not enjoy. Standing notation. Throughout this paper (X, T, wx) will denote a transformation group with compact Hausdorff phase space, G its transition group, and E its enveloping semigroup. If Q is a concept defined in terms of (X, T, 7r) and (Y, T, p) is another transformation group with phase group T and compact Hausdorff phase space Y, then Qy or Q(Y, T, p) will denote the same concept defined in terms of (Y, T, p). Thus Gy denotes the set [pt/tE T] and Ey denotes the closure of Gy in Yy. REMARK 1. Since XX may be regarded as the set of maps of X into X, a semigroup structure may be introduced into XX. Thus if p, qEXX, then pq denotes the map of X into X such that x(pq) = (xp)q (xEX). Provided with this structure and its cartesian product topology XX becomes a compact semigroup. The maps p--qp (pCXx) of XX into XX are continuous for all qCXX, and the maps p--pq (pEXx) of XX into XX are continuous for all continuous maps qEXx. Moreover E is a closed hence compact sub-semigroup of Xx. For proofs see [1 ] and [2 ]. REMARK 2. The map u: EXTE such that (p, t)a=pwt (pEE, tzT) defines a transformation group with phase space E and phase group T.