Algebra in superextensions of groups
by
Volodymyr Gavrylkiv
Precarpathian National University, Ukraine
Coauthors: Taras Banakh

In the talk we shall discuss the properties of the semigroups of maximal linked systems. By definition, a family L of subsets of a set X is called a linked system on X if A∩B is nonempty for all A, B ∈ L. Such a linked system is maximal linked if it coincides with any linked system M on X that contains L. For example, each ultrafilter is a maximal linked system. The space l(X) of all maximal linked systems on X is called the superextension of X, and is endowed with the topology generated by the sub-base consisting of the sets U⁺ = {L ∈ l(X):U ∈ L}, where U runs over subsets of X.

It is known that each binary operation * on X extends to a right topological operation on bX, the Stone-Cech compactification of X, playing a crucial role in Combinatorics of Numbers. In the same way the operation * can be further extended to a right-topological operation on l(X) by the formula: A*B={C ⊂ X: {x ∈ X: x^-1C ∈ B} ∈ A}. If the operation * on X is associative, then it extends to an associative operation on l(X). In this case bX is a subsemigroup of l(X). In the sequel G is a group.

We start with characterization the superextensions l(G) possessing (right) zeros.

Theorem 1 The superextension l(G) of a group G possesses a right zero if and only if G is odd in the sense that the order of each element of G is odd.

Theorem 2 The superextension l(G) has a left zero if and only if l(G) has a zero if and only if |G| ∈ {1, 3, 5}.

Theorem 3 The superextension l(G) of a group G is commutative if and only if |G| ≤ 4.

Next, we describe cancelative elements of the superextensions. Recall that an element x of a semigroup S is right cancelable if for every a, b ∈ X the equation x*a=b has at most one solution x ∈ S. We say that a maximal linked system L ∈ l(G) (i) has finite support if there is a finite family F ⊂ L of finite subsets of G such that each set L ∈ L contains a set F ∈ F; (ii) is free if for each L ∈ L and each finite subset F ⊂ G the complement L\F belongs to L.

Theorem 4 Let G be a group. A maximal linked system L ∈ l(G) is right cancelable in l(X) provided for every x ∈ X there is a set S_x ∈ L such that the family {x*S_x: x ∈ X} is disjoint.

Theorem 5 For each countable group G the subsemigroup l^○(G) of free maximal linked systems contains an open dense subset consisting of right cancelable elements in the semigroup l(G).

By definition, the topological center of a right-topological semigroup S is the set of all elements a ∈ S such that the left shift l_a:S→ S, l_a(x) = a*x, is continuous.

Theorem 6 For any countable group G the topological center of the semigroup l(G) coincides with the set l^•(G) consisting of all maximal linked systems with finite support.

Theorem 7 For any countable infinite group G the algebraic center of l(G) coincides with the algebraic center of X.

For finite groups this theorem is not true.

Remark 1 The semigroup l(G) contains a central element distinct from a principal ultrafilter if 3 ≤ |G| ≤ 5.

References:

T. Banakh, V. Gavrylkiv, O. Nykyforchyn. Algebra in superextensions of groups, I: zeros and commutativity, (preprint in arXiv:0802.1853).

T. Banakh, V. Gavrylkiv. Algebra in the superextensions of groups, II: cancelativity and centers, (preprint in arXiv:0802.1856).

T. Banakh, V. Gavrylkiv. Algebra in the superextensions of groups, III: minimal left ideals of l(Z), (prepint in arXiv:0805.1583)

T. Banakh, V. Gavrylkiv. Algebra in the superextensions of groups, IV: representation theory, (in preparation).

V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Matem. Studii (preprint in arXiv:0802.1859).

Date received: July 4, 2008