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Register a new userAlgebra in superextension of groups, II: cancelativity and centers
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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 $X$.
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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.
It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasovs question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
Hindmans celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Mengers classic covering property. The methods include, in addition to Hurewiczs game theoretic characterization of Mengers property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides stro
As proved by Dimov [Acta Math. Hungarica, 129 (2010), 314--349], there exists a duality L between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight and of dimension of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X)=w(L(X)), and if, in addition, X is normal, then dim(X)=dim(L(X)).
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.
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