On a topological simple Warne extension of a semigroup

In the paper we introduce topological $\mathbb{Z}$-Bruck-Reilly and topological $\mathbb{Z}$-Bruck extensions of (semi)topological monoids which are generalizations of topological Bruck-Reilly and topological Bruck extensions of (semi)topological monoids and study their topologizations. The sufficient conditions under which the topological $\mathbb{Z}$-Bruck-Reilly ($\mathbb{Z}$-Bruck) extension admits only the direct sum topology and conditions under which the direct sum topology can be coarsened are given. Also, topological characterizations of some classes of $I$-bisimple (semi)topological semigroups are given

Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse $\wedge$-semigroups arise as completions of inverse semigroups we call pre-Boolean. An inverse $\wedge$-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where the definition of a tight filter is obtained by combining work of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson-Higman groups $G_{n,r}$. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz-Krieger $C^{\ast}$-algebras.Comment: The presentation has been sharpened up and some minor errors correcte

Hypergroups and Hypergroup Algebras

The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous representations of topological hypergroup algebras.Comment: 52 page

Extreme Elements in Semigroups (Quasiorders, Green\u27s).

We study the elements of a semigroup which are minimal or maximal with respect to Green\u27s quasiorders. Part 1 begins with a preliminary review. The sets of minimal elements are characterized in terms of minimal ideals. We discuss the relationship between the min set of a semigroup and the min set of a subsemigroup. The sets of maximal elements are characterized, and it is shown that these sets do not necessarily satisfy any inclusion relationships to each other. We discuss the max sets of subsemigroups and product semigroups. Conditions are given under which the max sets and min sets can intersect. We define the concept of a paved semigroup and present conditions under which homomorphisms preserve sets of maximal elements. The translational hull is discussed in Part 2. We compare the condition that a semigroup S is H paved with the condition that S = ESE, where E is the set of idempotents of S. We prove that if S is a subsemilattice of a finite semilattice T and if their max sets are equal, then the degree of S is at most the degree of T. Topological results appear in Part 3. Extreme sets of compact semigroups are discussed. An example is given in which the set of nonmaximal elements can be extended in more than one way. We compare the max set with various topological notions of boundary. Part 4 contains results on divisibility and on the Nambooripad partial order. Conditions are given under which the minimal sets inherit divisibility properties of the semigroup. We prove that divisibility of a semigroup very strongly implies divisibility of its max sets. Finally, we show that any element of a regular semigroup which is maximal with respect to the H, R, or L quasiorder is maximal with respect to the Nambooripad partial order

Profinite Groups Associated to Sofic Shifts are Free

We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the \J-class associated to an irreducible sofic shift