Topological invariance of ideals in mobs

Abstract

A mob is a Hausdorff space together with a continuous associative multiplication. In all that follows S will be a compact mob. A set TCS is a left ideal if TpD and if STCT. It is clear how to define right ideal and (two-sided) ideal. Numakura [7] has shown that S contains minimal ideals of all three sorts and these are closed sets. We let K be the minimal ideal of S. Improving some results of [11], we show among other things that, with additional assumptions on S, it is possible to give a completely topological definition of K. It will be seen also that if N is a sufficiently large subgroup of S, then the cohomology structure of S is the same as that of N. This will be done by showing that N= K. From this it follows that N is a homomorphic retract of S. But N need not be a deformation retract of S, see [3 ]. The Alexander-Kolmogoroff cohomology group of the space X with coefficient group G will be denoted by Hn(X, G), Spanier [8]. We sometimes write Hn(X) for Hn(X, G). It is possible to define a dimension function (Haskell Cohen [4]) by letting cd(X, G)<?n if the natural homomorphism Hn(X, G) into Hn(A, G) is onto for each closed A CX. If X is compact Hausdorff then cd(X, integers) is the covering dimension, [1] and [5]. Cohen [4] showed that cd(X, G) cannot exceed the covering dimension for a compact X. If hCHn(X), then hi A will denote the image of h in Hn(A) under the natural homomorphism, A CX. A continuum is a compact connected Hausdorff space.