Algebra in the superextensions of groups
by
Volodymyr Gavrylkiv
Precarpathian National University
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.

Finally, given an Abelian group G we desribe the structure of minimal left ideals of the superextension l(G).

By C_2^k={z ∈ C:z^2k=1} we denote the cyclic group of order 2^k. Let also C_2^∞=∪_k=1^∞ C_2^k be the quasi-cyclic 2-group.

For a group G by q(G, C_2^k) we denote the number of normal subgroups H ⊂ G with quotient G/H isomorphic to C_2^k. It is easy to see that for k ∈ N

q(G, C2k)= h(G, C2k)-h(G, C2k-1) 2k-1

where h(G, C_2^k) is the number of homomorphisms from G into C_2^k.

Theorem 8 For an Abelian group G and an idempotent e in the minimal ideal of l(G) the following conditions are equivalent:

1. q(G, C_2^∞)=0.

2. All the minimal left ideals are topological semigroups.

3. Some maximal subgroup of the minimal ideal of l(G) is compact.

4. All maximal subgroups of the minimal ideal of l(G) are topological groups.

5. The maximal subgroup H(e)=e·l(G)·e is topologically isomorphic to the Tychonov product ∏_k=1^∞ (C_2^k)^{q(G, C_2k)}.

6. The set of idempotents of any minimal left ideal of l(G) is compact.

7. The set E(l(G)·e) of idempotents of the minimal left ideal l(G)·e is a compact semigroup of left zeros, homeomorphic to the cube {0, 1}^l with
   

   l = ∞ å k=1  q(G, C2k)·((k+1)22k-1-k-k).

8. The minimal left ideal l(G)·e is topologically isomorphic to the product H(e)×E(l(G)·e).

9. The continuous homomorphism l(q):l(G)→ l(G₂) induced by the pro-2-group reflexion q:G→ G₂ is injective on each minimal left ideal of l(G).

The last item of this theorem requires some explanations. The pro-2-group reflexion q:G→ G₂ is defind as follows. Consider the family D of normal subgroups H of G with |G/H|=2^k for some k ∈ N. The quotient homomorphisms q_H:G→ G/H, H ∈ D, compose a homomorphism q:G→∏_{H ∈ D}G/H to the Tychonov product of finite 2-groups. The closure of q(G) in ∏_{H ∈ D}G/H is denoted by G₂ and the homomorphism q:G→ G₂ is called the pro-2-group reflexion of G.

For example, the pro-2-group reflexion Z₂ of the group Z of integer is the group of integer 2-adic numbers.

Since the compact group G₂ is the inverse limit of finite 2-groups G/H, H ∈ D, its superextension l(G₂) in the inverse limit of the finite semigroups l(G/H), H ∈ D. Consequently, l(G₂) is a compact zero-dimensional topological semigroup. Now it is clear that the injectivity of the homomorphism l(q):l(G)→l(G₂) on a minimal left ideal of l(G) implies that this ideal is topologically isomorphic to a minimal left ideal in l(G₂) and thus is a topological semigroup.

References:

T. Banakh, V. Gavrylkiv, O. Nykyforchyn. Algebra in superextensions of groups, I: zeros and commutativity, Matem. Studii (submitted).

T. Banakh, V. Gavrylkiv. Algebra in the superextensions of groups, II: cancelativity and centers, Algebra and Discrete Math. (submitted).

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

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 (to appear).

Date received: February 28, 2008