Algebra in the superextensions of twinic groups

Abstract

Given a semilattice X we study the algebraic properties of the semigroup υ(X) of upfamilies on X.The semigroup υ(X) contains the Stone-Čech extension β(X), the superextension λ(X), and the space of filters ϕ(X) on X as closed subsemigroups. We prove that υ(X) is a semilattice iff λ(X) is a semilattice iff ϕ(X) is a semilattice iff the semilattice X is finite and linearly ordered. We prove that the semigroup β(X) is a band if and only if X has no infinite antichains, and the semigroup λ(X) is commutative if and only if X is a bush with finite branches. IntroductionOne of powerful tools in the modern Combinatorics of Numbers is the method of ultrafilters based on the fact that each (associative) binary operation * : X × X → X defined on a discrete topological space X extends to a right-topological (associative) operation * : [11]. The Stone-Čech extension β(X) is the space of ultrafilters on X. The extension of the operation from X to β(X) can be defined by the simple formula:Endowed with the so-extended operation, the Stone-Čech compactification β(X) becomes a compact righttopological semigroup. The algebraic properties of this semigroup (for example, the existence of idempotents or minimal left ideals) have important consequences in combinatorics of numbers, see [9], [11].In [8] it was observed that the binary operation * extends not only to β(X) but also to the space υ(X) of all upfamilies on X. By definition, a family F of non-empty subsets of a discrete space X is called an upfamily if for any sets A ⊂ B ⊂ X the inclusion A ∈ F implies B ∈ F . The space υ(X) is a closed subspace of the double power-set P(P(X)) endowed with the compact Hausdorff topology of the Tychonoff power {0, 1} P(X) . In the papers [7], [8], [1]-[4] the space υ(X) was denoted by G(X) and its elements were called inclusion hyperspaces 1 .The extension of a binary operation * from X to υ(X) can be defined in the same way as for ultrafilters, i.e., by the formula (1) applied to any two upfamilies U, V ∈ υ(X). If X is a semigroup, then υ(X) is a compact Hausdorff right-topological semigroup containing β(X) as closed subsemigroups. The algebraic properties of this semigroups were studied in details in [8].The space υ(X) of upfamilies over a discrete space X contains many interesting subspaces. First we recall some definitions. An upfamily A ∈ υ(X) is defined to be • a filter if A 1 ∩ A 2 ∈ A for all sets A 1 , A 2 ∈ A;• an ultrafilter if A = A ′ for any filter A ′ ∈ υ(X) containing A;• linked if A ∩ B = ∅ for any sets A, B ∈ A;• maximal linked if A = A ′ for any linked upfamily A ′ ∈ υ(X) containing A.1991 Mathematics Subject Classification. 06A12, 20M10. Key words and phrases. semilattice, band, commutative semigroup, the space of upfamilies, the space of filters, the space of maximal linked systems, superextension. 1 We decided to change the terminology and notation after discovering the paper [12, 2.7.4] that discusses monadic properties of the up-set functor υ.

On infinite groups in which all abelian subgroups are locally cyclic

“…Moreover Q 2 ∞ is locally nilpotent but not nilpotent, and every element of Q 2 ∞ \C 2 ∞ has order 4. Let C t denote the cyclic group of order t ∈ N. It is well known that a 2-group has an unique element of order 2 if and only if it is isomorphic to C 2 n or Q 2 n for some n ∈ N ∪ {∞} (see, for instance, [1,Theorem 8.1]). It easily follows that the infinite quaternion group is anticommutative.…”

Bases in finite groups of small order

“…Difference bases and difference characteristics in dihedral and Abelian groups were investigated in [3,4]. Difference bases have applications in the study of structure of superextensions of groups, see [1,5,9].…”

On the automorphism group of the superextension of a semigroup

“…The superextensions of semilattices were studied in [4]. In particular, it was shown that λ(L 3 ) ∼ = L 4 , and hence Aut (λ(L 3 )) ∼ = Aut (L 4 ) ∼ = C 1 .…”