Lattice of compactifications of a topological group

We show that the lattice of compactifications of a topological group G is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification bG of G. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.


Introduction
Let X be a Tychonoff space.It is well known that the collection K(X) of all Hausdorff compactifications of X forms a complete upper semi-lattice under the order relation defined by c 1 X ≤ c 2 X if and only if there is a continuous map f : c 2 X → c 1 X, which leaves X pointwise fixed.In general K(X) is not a complete lattice and it is a complete lattice if and only if X is locally compact.There are many results studying the relationship between topological properties of X and the order structure of K(X) (see [4], [9]- [13], [16]).
Here we consider the Hausdorff compactfications of a Hausdorff topological group G.
Let G be a topological group.Recall that a compactification of G is a pair (K, ϕ), where K is a compact Hausdorff group and ϕ : G → K is a dense continuous homomorphism.For two compactifications (K, ϕ) and (H, ψ) of G, we say that (K, ϕ) and (H, ψ) are equivalent if there is a topological isomorphism h : K → H such that ψ = h • ϕ.This defines an equivalence relation on the collection of all Hausdorff compactifications of G.We use the same symbol K(G) (like in the case of topological spaces) to denote the set of all equivalence classes of the Hausdorff compactifications of G.There exists a natural order relation on K(G) defined as follows: (K, ϕ) ≤ (H, ψ) if there exists a continuous homomorphism µ : The following result is well known.For convenience of the readers we give a proof here.
Proof.The necessity is clear.We only need to show the sufficiency.Suppose that (K, ϕ) ≤ (H, ψ) and (H, ψ) ≤ (K, ϕ) both hold.We have continuous homomorphisms f : H → K and g : By this lemma, (K(G), ≤) is a partially ordered set.Clearly K(G) has the least element, that is the trivial group and trivial homomorphism, and also has the largest element, the Bohr compactification of G.
In this note we investigate the order algebraic structure of the compactification lattice K(G) of a given topological group G and its relationship with the properties of G.

Preliminaries
Throughout this paper we consider the category TopGrp of Hausdorff topological groups and continuous homomorphisms.TopAb and Ab respec-tively denote the category of topological abelian groups and the category of discrete abelian groups.The categories of complete lattice and suppreserving maps and the category of complete lattice and inf-preserving maps are denoted by CLat and CLat op , respectively.
We denote by N the set of positive natural numbers; by Z the integers, by Q the rationals, by R the reals, and by T the unit circle group which is identified with R/Z.We write U = ∞ n=1 U (n), where U (n) is the group of all n×n unitary matrices in GL n (C).According to Peter-Weyl's theorem, for every compact group K, the continuous homomorphisms K → U separate the points of K [7].It follows that every compact group can be topologically embedded into some power of U. Hence we can take U in TopGrp in the place of the circle group T in the category TopAb of topological Abelian groups.The cyclic group of order n > 1 is denoted by Z(n).
The subgroup generated by a subset X of a group G is denoted by X , and x is the cyclic subgroup of G generated by an element x ∈ G.The abbreviation K ≤ G means that K is a subgroup of G, and N ¡ G means that N is a normal subgroup of G.
For an arbitrary topological group G, we denote the family of all continuous homomorphisms from G to U by the symbol C * (G).The Bohr compactification of G is denoted by b : G → bG.The group G endowed with the Bohr topology, that is, the topology induced by the family of all continuous homomorphisms from G to U, is denoted by Throughout the paper all topological groups are assumed to be Hausdorff.All unexplained topological terms can be found in [3].
3 Compactification lattice of a topological group Lemma 3.1.For every topological group G, the partially ordered set K(G) is a complete lattice.
Proof.We only need to show that every family of compactifications of G has supremum. Let the closure of the image of G under c i , and c : G → cG the restriction of the mapping c i .Then c : G → cG is a compactification of G, and the equalities Indeed, suppose that k : G → K is a compactification of G such that for each i ∈ I, there exists a continuous homomorphism Let also CN (bG) be the set of all closed normal subgroups of bG ordered by inverse inclusion.We have the following result.Proposition 3.2.L : K(G) → CN (bG) is an isomorphism.
Proof.Notice that L has an inverse map which sends each closed normal subgroup N ¡ G to the compactification q • b : G → bG/N , where q : bG → bG/N is the quotient map.Also for two compactifications c 1 : Let G and H be topological groups.If bG is topologically isomorphic to bH, then K(G) and K(H) are isomorphic.
A classical result in [11] showed that if X and Y are locally compact spaces, then their lattices of compactifications K(X) and K(Y ) are isomorphic if and only if βX \ X and βY \ Y are homeomorphic.For a topological group G, the remainder bG \ b(G) in general does not determine the order structure of K(G).Indeed, if we take S(X) the permutation group of an infinite set X, we know that S(X) is a minimally almost periodic groups, that is, N (S(X)) = S(X).Put H = T the unit circle group.Then bS(X) is the trivial group and bH = H.Hence the remainder bS Corollary 3.4.Let L be a complete lattice.L is isomorphic to the compactification lattice K(G) of a topological group G if and only if L is isomorphic to the lattice of all closed normal subgroups of a compact group.
Let G be a topological group and let H ≤ G be a dense subgroup of G. Then for every compactification c : G → cG of G the restriction of c to H is a compactification of H. Conversely every compactification c : H → cH of H admits a unique extension c : G → cH.Hence the following result is clear.
If G is a topological Abelian group, we can give an alternative description of K(G).Let C * (G) denote the group of all continuous characters on G, that is, the group of all continuous homomorphisms f : G → T. Suppose that N ≤ C * (G).Then we have a compactification c N : where c N G and c M G are the compactifications corresponding to N and M , respectively.
Proposition 3.6.The partially ordered set (Subgp(C * (G))/ ∼, ≤) is isomorphic to K(G).Note that the above corollary is true for an arbitrary topological group G, though then C * (G) is not a group.
Let L be a complete lattice.An element a ∈ L is called compact if for every family {a i | i ∈ I} ⊂ L with a ≤ a i , there exists finite set J ⊂ I such that a ≤ i∈J a i .Let k(L) be the set of all compact elements of L. A complete lattice L is said to be an algebraic lattice if a = {c ∈ k(L) | c ≤ a} for every a ∈ L. L is said to be a dual algebraic lattice if the dual lattice L op of L is algebraic.Theorem 3.8.If L is a compactification lattice of a topological Abelian group, then there is a dual algebraic lattice S such that L is a retraction of S.
Proof.Suppose that G is a topological Abelian group and L is order isomorphic to the compactifiction lattice K(G) of G. Let S be the lattice of all subgroups of bG.We first show that S is a dual algebraic lattice (with respect to the inverse inclusion order).
Indeed, S is a complete lattice and it contains L as a sub-upper semilattice.It is clear that the family of all finitely generated subgroups of bG and the family of all compact element of S op coincide, and also every subgroup H ≤ bG can be represented as a supremum of a family of finitely generated subgroups of bG.
Let r : S → L be such that for every a ∈ S, r(a) is in the closure of a in bG.Then r is a retraction.

The contravariant lattice-valued functor
Let f : G → H be a continuous homomorphism.We define K(f ) : K(G) → K(H) as the mapping which sends every compactification c :  Proof.For two continuous homomorphisms f : G → H and g : H → M of topological groups, it is clear that K(g • f ) = K(g) • K(f ).We only need to show that K(f ) : CN (bH) → CN (bG) preserves arbitrary joins, for any continuous homomorphism f : G → H. K(f ) maps H to G, that is, it preserves the bottom element.Let {N i | i ∈ I} be a family of closed normal subgroups of bH.We have i∈I N i = i∈I N i , and hence This shows that K(f ) preserves arbitrary joins.Lemma 4.2.Let G and H be topological groups and let f : G → H be a dense continuous homomorphism.Then K(f We know that in the category TopAb of topological abelian groups, monomorphisms are precisely one-to-one continuous homomorphisms and epimorphisms are precisely dense continuous homomorphisms.Hence we have the following result.Let G be a topological group.We know that the Bohr compactification bG of G can be obtained in two steps: first taking pG = (G/N (G)) + the quotient group G/N (G) endowed with the Bohr topology.Let q : G → pG be the quotient map, which is in fact the precompact reflection of G. Next we put bG = pG to be the Raȋkov completion of pG.Then bG is the Bohr compactification of G. Suppose that f : G → H is a continuous homomorphism.Then f has a continuous homomorphism extension pf : pG → pH such that the following diagram commutes: Proof.Suppose that N (G) = f − (N (H)).By the above argument, ker(bf ) is trivial, hence bf is one-to-one.It follows that bf : bG → bH is a closed embedding, thus K(f is surjective, then there exists a closed normal subgroup N ¡ bH such that bf − (N ) = {e bG }.It follows that ker(bf ) = {e bG }.This implies that N (G) = f − (N (H)).
Let G and H be two discrete abelian groups and let f : G → H be a one-to-one homomorphism.Then each homomorphism g : G → T has an extension ĝ : In general the functor K : TopAb op → CLat does not preserve epimorphisms.Indeed, if we take a topological Abelian group H such that H is not maximally almost periodic and denote by G the group H endowed with discrete topology, then N (G) is the trivial group and N (H) = {e H }. By Lemma 3.4, K(id) : K(G) → K(H) is not surjective.Proposition 4.6.If f : G → H is a dense continuous homomorphism such that N (G) = f − (N (H)), then bG and bH are topologically isomorphic.So K(G) is order isomorphic to K(H).
Proof.Clearly we have ker(bf is injective if and only if for any two closed subgroups N 1 and N 2 of bH, N 1 ∩ bG = N 2 ∩ bG implies that N 1 = N 2 . The functor K : TopGp op → CLat does not preserve coproducts.Indeed, if we take G = H = Z(2), then bG = bH = Z(2), and b(G × H) = Z(2) × Z(2).As | K(G × H) |= 5 and | K(G) × K(H) |= 4, so K(G × H) is not isomorphic to K(G) × K(H).Since in the category CLat of complete lattices, coproducts of objects are equivalent to products of objects, hence K(G × H) is not the coproduct of K(G) and K(H).
The following result was proved in [6].
Lemma 4.8.Let {G i | i ∈ I} be a family of topological groups.Then b i∈I G i is topologically isomorphic to i∈I bG i .
Proposition 4.9.Let {G i | i ∈ I} be a family of topological groups.Then K(G i ) is a sub-complete lattice of the complete lattice K( G i ), and there exists an inf-preserving mapping r : K( G i ) → K(G i ) which leaves K(G i ) pointwise fixed.
It is a natural question whether for two topological abelian groups G and H, K(G) isomorphic to K(H) implies that bG and bH are topological isomorphic.But in general this is not true.For example, we take G = Z(2) and H = Z(3), then K(G) = K(H) are two-elements lattices, but bG = G, bH = H.Open Question 4.12.Let G and H be topological groups.Can one give sufficient and necessary conditions on G and H for K(G) to be isomorphic to K(H)?
Let G be a topological group.For every compactification c : G → cG of G, c can be uniquely factored through b : G → bG as c = c b • b: Clearly, c b is surjective since c is dense and c b is closed.Denote by L(cG) = ker(c b ) the kernel of c b .Then L(cG) is a closed normal subgroup of bG.
Equivalently, if we regard K(G) as CN (bG), and K(H) as CN (bH), then K(f ) : CN (bH) → CN (bG) sends every closed normal subgroup N ¡bH to the closed normal subgroup bf − (N )¡bG, where bf : bG → bH is the extension of f .

Proposition 4 . 1 .
The assignment G → K(G) is a contravariant functor from the category TopGp op to the category CLat of complete lattices.
It is clear thatKer(pf ) = {e pG } if and only if N (G) = f − (N (H)).By the construction of the Raȋkov completion, we know that Ker(pf ) = {e pG } if and only if Ker(bf ) = {e bG }: dense.It follows that bf : bG → bH is dense and surjective, since it is closed.Furthermore, we have the following result.Proposition 4.7.Let f : G → H be a continuous homomorphism.Then K(f ) : K(H) → K(G) is an isomorphism if and only if bf : bG → bH is an embedding and for any two closed subgroups N 1 and N 2 of bH, N 1