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Register a new userCharacterizing compact Clifford semigroups that embed into convolution and functor-semigroups
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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 $G$).
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Let $mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Yin mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $mathcal C$-$closed$ if for each congruence $approx$ on $X$ the quotient semigroup $X/_approx$ is $mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $Isubseteq X$ there are elements $x,yin I$ such that $xy otin{x,y}$. We prove that a semigroup $X$ is $mathcal C$-closed if it admits a homomorphism $h:Xto E$ to a chain-finite semilattice $E$ such that for every $ein E$ the semigroup $h^{-1}(e)$ is $mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $Asubseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $Xsetminus H(X)$.
We find necessary and sufficient conditions on an (inverse) semigroup $X$ under which its semigroups of maximal linked systems $lambda(X)$, filters $phi(X)$, linked upfamilies $N_2(X)$, and upfamilies $upsilon(X)$ are inverse.
We present a survey of results on profinite semigroups and their link with symbolic dynamics. We develop a series of results, mostly due to Almeida and Costa and we also include some original results on the Schutzenberger groups associated to a uniformly recurrent set.
We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.
It is shown that if V is a local monoidal pseudovariety of semigroups, then K(m)V, D(m)V and LI(m)V are local. Other operators of the form Z(m)(_) are considered. In the process, results about the interplay between operators Z(m)(_) and (_)*D_k are obtained.
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