Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

Abstract

We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P (G) over some topological group G if and only if S embeds into the semigroup exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F (G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F (G) contains a G-invariant element (which is an analogue of the Haar measure on G).

Extending binary operations to funtor-spaces

“…Assume conversely that (xy)z = x(yz) for some points x, y, z ∈ X. Since X is Hausdorff, the points (xy This problem was addressed in [10], [11] for the monadic functor G of inclusion hyperspaces, in [2]- [5] for the functor of superextension λ, in [1], [12], [15] for the functor P of probability measures and in [6], [7], [8], [18] for the hyperspace functor exp.…”