We characterize semigroups $X$ whose semigroups of filters $varphi(X)$, maximal linked systems $lambda(X)$, linked upfamilies $N_2(X)$, and upfamilies $upsilon(X)$ are commutative.
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.
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.
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 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 constr
uct 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 continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subs
emigroup of all elements of TX that have finite support.
We show that for any discrete semigroup $X$ the semigroup operation can be extended to a right-topological semigroup operation on the space $G(X)$ of inclusion hyperspaces on $X$. We detect some important subsemigroups of $G(X)$, study the minimal id
eal, the (topological) center, left cancelable elements of $G(X)$, and describe the structure of the semigroups $G(IZ_n)$ for small numbers $n$.
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 $X$.
