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Register a new userAlgebra in superextensions of inverse semigroups
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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.
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Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $lambda(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators.
Given a semilattice $X$ we study the algebraic properties of the semigroup $upsilon(X)$ of upfamilies on $X$. The semigroup $upsilon(X)$ contains the Stone-Cech extension $beta(X)$, the superextension $lambda(X)$, and the space of filters $phi(X)$ on $X$ as closed subsemigroups. We prove that $upsilon(X)$ is a semilattice iff $lambda(X)$ is a semilattice iff $phi(X)$ is a semilattice iff the semilattice $X$ is finite and linearly ordered. We prove that the semigroup $beta(X)$ is a band if and only if $X$ has no infinite antichains, and the semigroup $lambda(X)$ is commutative if and only if $X$ is a bush with finite branches.
A family $mathcal L$ of subsets of a set $X$ is called linked if $Acap B eemptyset$ for any $A,Binmathcal L$. A linked family $mathcal M$ of subsets of $X$ is maximal linked if $mathcal M$ coincides with each linked family $mathcal L$ on $X$ that contains $mathcal M$. The superextension $lambda(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : Xtimes X to X$ can be extended to an associative binary operation $*: lambda(X)timeslambda(X)tolambda(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality $leq 5$.
As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.
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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