Characterizing Compact Clifford Semigroups That Embed into Convolution and Functor-semigroups

We study algebraic and topological properties of the convolution semigroups of probability measures on 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).