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

摘要

We study algebraic and topological properties of the convolution semigroupsof probability measures on a topological groups and show that a compactClifford 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 semigroupand has zero-dimensional maximal semilattice. We also show that such a Cliffordsemigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compacttopological group $G$ for each weakly normal monadic functor $F$ in thecategory of compacta such that $F(G)$ contains a $G$-invariant element (whichis an analogue of the Haar measure on $G$).