A Semigroup Associated With a Transformation Group
Robert Richmond Ellis · Transactions of the American Mathematical Society · 1960
Let (X, T, ir) be a transformation group with compact Hausdorff phase space X, and let G= [ir'/tET] 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, vr) is by definition the closure of G in Xx [2].In the first half of this paper algebraic properties of £ 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 £.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 £ does not enjoy.Standing notation.Throughout this paper (X, T, it) will denote a transformation group with compact Hausdorff phase space, G its transition group, and £ its enveloping semigroup.If Q is a concept defined in terms of (X, T, it) and (F, 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 (F, T, p).Thus Gy denotes the set [p'/tET] 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-> £ such that (p, t)a = pw' (PEE, tET) defines a transformation group with phase space £ and phase group T.