Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups
We study algebraic and topological properties of the convolution semigroup 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).
KeywordsConvolution semigroup Global semigroup Hypersemigroup Clifford semigroup Regular semigroup Topological group Radon measure Weakly normal monadic functor
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