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

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### 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 $$G$$).

### Most cited references5

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### Universal Objects in Some Classes of Clifford Topological Inverse Semigroups

(2007)
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### Functors of probability measures in topological categories

(1998)
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### Idempotent measures on semigroups

(1962)
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### Author and article information

###### Journal
06 November 2008
2011-08-02
###### Article
10.1007/s00233-011-9319-5
0811.1026