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Register a new userAlgebra in the superextensions of groups: minimal left ideals
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We prove that the minimal left ideals of the superextension $lambda(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $lambda(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers.
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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.
Given a countable group $X$ we study the algebraic structure of its superextension $lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$mathcal Acircmathcal B={Csubset X:{xin X:x^{-1}Cinmathcal B}inmathcal A}$$ that extends the group operation of $X$. We show that the subsemigroup $lambda^circ(X)$ of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of $lambda(X)$ coincides with the subsemigroup $lambda^bullet(X)$ of all maximal linked systems with finite support. This result is applied to show that the algebraic center of $lambda(X)$ coincides with the algebraic center of $X$ provided $X$ is countably infinite. On the other hand, for finite groups $X$ of order $3le|X|le5$ the algebraic center of $lambda(X)$ is strictly larger than the algebraic center of $X$.
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.
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.
The superextension $lambda(X)$ of a set $X$ 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 isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $leq 5$.
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