On topological Clifford semigroups embeddable into products of cones over topological groups

In this paper we give characterizations of topological Clifford semigroups which are embeddable into Tychonoff products of topological semilattices and cones over topological groups. Also we characterize topological Clifford semigroups which embed into compact topological Clifford semigroups.


Introduction
This paper was motivated by the problem of recognizing topological Clifford semigroups which can be embedded into compact topological Clifford semigroups. We shall resolve this problem for topological Clifford U -semigroups (i.e., topological Clifford semigroups whose idempotent band E is a U -semilattice). Topological U -semilattices and their relation to other classes of topological semilattices will be discussed in Section 2.4. In Theorem 5.1 we shall prove that a topological Clifford U -semigroup S embeds into a compact topological Clifford semigroup if and only if S is ditopological, the maximal semilattice E of S embeds into a compact topological semilattice and each maximal subgroup H e , e ∈ E, of S embeds into a compact topological group. Ditopological Clifford semigroups were introduced and studied in [3]. The class of such semigroups contains all compact topological Clifford semigroups, all topological semilattices, all topological groups and is closed under many operations over topological Clifford semigroups. We shall discuss ditopological Clifford semigroups in Subsection 2.7. In Section 3 we shall prove that each ditopological Clifford U -semigroup S embeds into a Tychonoff product of topological semilattices and cones over maximal subgroups H e , e ∈ E, of S.

Preliminaries
In this section we recall some definitions and prove some auxiliary results.

Topological spaces
All topological spaces considered in this paper are Hausdorff. A topological space X is called • zero-dimensional if closed-and-open subsets form a base of the topology of X ; • punctiform if each non-empty compact connected subset of X is a singleton. It is clear that each zero-dimensional topological space is punctiform. By [6, 1.4.5], a locally compact space is zero-dimensional if and only if it is punctiform.
The weight w(X ) of a topological space X is the smallest cardinality |B| of a base B of the topology of X .

Semigroups
A semigroup is a set S endowed with an associative binary operation * : S × S → S. In the sequel we shall often omit the symbol * of the operation and write x y instead of x * y.
A semigroup S is called • a semilattice if x y = yx and x x = x for all x, y ∈ S; • regular if for each x ∈ X there is y ∈ X such that x yx = x and yx y = y; • an inverse semigroup if for each x ∈ X there exists a unique element x −1 ∈ S such that x x −1 x = x and x −1 x x −1 = x −1 ; • a Clifford semigroup if S is inverse and x x −1 = x −1 x for all x ∈ X . It is well-known [7, 5.1.1] that a semigroup S is inverse if and only if S is regular and the set E = {e ∈ S : ee = e} is a commutative subsemigroup of S. In this case E will be called the maximal semilattice of S. If S is a Clifford semigroup, then the map is a semigroup homomorphism, and for every idempotent e ∈ E the preimage π −1 (e) coincides with the maximal subgroup H e = {x ∈ S : x x −1 = e = x −1 x} of S containing the idempotent e.
A subset I of a semigroup S is called an ideal if S I S ⊂ I .

Topological semigroups
By a topological (inverse) semigroup we understand an (inverse) semigroup S endowed with a topology such that the semigroup operation * : S × S → S is continuous (and the inversion (·) −1 : S → S is continuous). If the inverse semigroup S is Clifford, then S will be called a topological Clifford semigroup.

Proposition 2.1 A compact topological semigroup S is Clifford if and only if S contains a dense Clifford subsemigroup.
Proof Let X ⊂ S be a dense Clifford subsemigroup of S. Let E = {e ∈ S : ee = e} be the set of idempotents in S and E X = X ∩ E. By the commutativity of the semigroup E X and the continuity of the semigroup operation * : S × S → S, the closureĒ X of E X in S is a semilattice.
Since the semigroup X is Clifford, for every x ∈ X there is a unique element Let pr 1 : S×S → S, pr 1 : (x, y) → x, be the coordinate projection. The projection pr 1 (D X ) ⊃ X , being a dense closed subset of S, coincides with S. Consequently, for each point x ∈ S there is a point x ∈ S such that (x, x ) ∈D X and hence x x x = x, x x x = x and x x = x x. This means that the semigroup S is regular.
We claim that E =Ē X . Given any idempotent e ∈ E, find an element e ∈ S such that (e, e ) ∈D X . Taking into account that ee e = e, e ee = e and ee = e e, we conclude that e = ee = (ee e)(ee e) = (eee )(eee ) = (ee )(ee ) = ee ∈ * (D X ) ⊂Ē X . Therefore, E =Ē X is a commutative semigroup. Being a regular semigroup with commuting idempotents, the semigroup S is inverse. To show that S is Clifford, it suffices to show that the set E =Ē X is contained in the center Z (S) = {z ∈ S : ∀x ∈ S xz = zx} of S. Observe that for every x ∈ X the set Z x = {z ∈ S : zx = xz} is closed in S and contains the set E X . Then E =Ē X ⊂ Z x for all x ∈ X . Now observe that for every e ∈ E the set Z e = {x ∈ X : xe = ex} is closed and contains the dense subset X of S. Consequently, Z e = S for all e ∈ E and hence E is contained in the center of S. By [11,II.2.6], the inverse semigroup S is Clifford.

Problem 2.2 Assume that a compact topological semigroup S contains a dense inverse subsemigroup. Is S inverse?
We shall say that a topological semigroup X embeds into a topological semigroup Y if there exists a semigroup homomorphism h : X → Y , which is a topological embedding. In this case we shall write X → Y .
A topological semigroup S is defined to be precompact if S embeds into a compact topological semigroup.

Proposition 2.3 A topological Clifford semigroup S is precompact if and only if S embeds into a compact topological Clifford semigroup K having weight w(K ) = w(S).
Proof The "if" part is trivial. To prove the "only if" part, assume that S is precompact and hence embeds into a compact topological semigroup K . We lose no generality assuming that S ⊂ K . By Theorem 2.29 [4], the compact topological semigroup K embeds into a Tychonoff product α∈A K α of metrizable topological semigroups K α , α ∈ A. By a standard argument one can show that there is a subset B ⊂ A of cardinality |B| ≤ w(S) such that the projection pr B : S → α∈B K α is a topological embedding. Then K B = α∈B K α is a compact topological semigroup of weight w(K B ) ≤ |B| ≤ w(S) containing a topological copy of S. Without loss of generality we can assume that S ⊂ K B . By Proposition 2.1, the closureS of S in K B is a Clifford semigroup. By Koch-Wallace-Kruming Theorem [9,10] the inversion ( ) −1 :S →S is continuous, which means thatS is a compact topological Clifford semigroup.
In Section 5 we shall consider the problem of detecting precompact topological Clifford semigroups in more details.
Let X, Y be two topological semigroups. By Hom(X, Y ) we denote the set of all continuous semigroup homomorphisms from X to Y . We shall say that X is is injective (resp. a topological embedding).
If a topological semigroup Y is precompact, then so is each Y -embeddable semigroup X .

Topological semilattices
A topological semilattice is a topological space E endowed with a continuous semilattice operation. An important example of a topological semilattice is the unit interval [0, 1] endowed with the usual topology and the operation of minimum (x, y) → min{x, y}. We denote this topological semilattice by I. The topological semilattice I contains a two-element subsemilattice 2 = {0, 1}.
Each topological semilattice E carries a partial order defined by x ≤ y if x y = x = yx. For a point x ∈ E let ↓x = {y ∈ E : y ≤ x} and ↑x = {y ∈ E : x ≤ y} be the lower and upper cones of x, respectively. For a subset A of a semilattice E we put ↓A = a∈A ↓a and ↑A = a∈A ↑a. A subset A ⊂ E will be called upper if A = ↑A.
Given two points x, y of a topological semilattice E, we write x y if y ∈ Int(↑x). For a point x ∈ E consider the sets ⇑x = {y ∈ E : x y} and ⇓x = {y ∈ E : y x}, and observe that ⇑x = Int(↑x).
A point x of a topological semilattice E will be called locally minimal if its upper cone ↑x is open in S. In this case ⇑x = ↑x. Observe that a point x ∈ E is locally minimal if and only if it is isolated in its lower cone ↓x.
A topological semilattice E is called These properties of topological semilattices relate as follows: Non-trivial implications from this diagram are proved in the following proposition.

Proposition 2.4 Let E be a topological semilattice.
(1) If E is locally compact and punctiform, then E is Lawson.
(2) If E is lower locally compact and punctiform, then E is a U 0 -semilattice; (

3) If E is lower locally compact and Lawson, then E is a U -semilattice; (4) If E is a U -semilattice, then E is
Proof Fix a countable base B of the topology of the space E. For every basic set B ∈ B we shall construct a countable set A B ⊂ B such that for any point x ∈ B there is a point a ∈ A B with x ∈ ⇑a.
Since E is a U -semilattice, for every point x ∈ B, there are a point a x ∈ B with x ∈ ⇑a x and a basic neighborhood U x ∈ B of x such that U x ⊂ ⇑a x . The family U B = {U x : x ∈ B} ⊂ B is countable and hence can be enumerated as U B = {V n : n ∈ ω}. For every n ∈ ω find a point x n ∈ B such that U n = U x n and observe that A B = {a x n : n ∈ ω} is a countable set with the required property: for any point x ∈ B there is a point a = a x n ∈ A B such that x ∈ U x = V n = U x n ⊂ ⇑a.
Then the countable union A = B∈B A B is a countable U -dense subset in E, which implies that the semilattice E is U -separable.

Topological Clifford U -semigroups
A topological Clifford semigroup S is defined to be a topological Clifford U -semigroup if its maximal semilattice E = {x ∈ S : x x = x} is a U -semilattice.

Reduced products over topological semigroups
Let E and H be topological semigroups and I be a closed ideal in E.
By the reduced product E × I H of E and H over the ideal I we mean the set I ∪ ((E \ I ) × H ) endowed with the smallest topology such that We shall be especially interested in reduced products of the form E × I e H where H is a topological group, E is a topological semilattice, e is an idempotent in E, and I e = E \ ⇑e is the closed ideal in E determined by the idempotent e. Here ⇑e = Int(↑e) is the interior of the upper cone of e in E.
For the compact topological semilattice I = [0, 1] and the idempotent e = 0 the ideal I e coincides with the singleton {0}. The reduced product I × {0} G will be denoted by G and called the cone over the topological group G. The cone G = I× {0} G contains the reduced product 2 × {0} G, which will be denoted byĠ and called the 0-extension of G. Observe thatĜ andĠ are topological Clifford U -semigroups.

Ditopological Clifford semigroups
For two subsets A, B of a semigroup S let Let S be a topological Clifford semigroup and π : S → E, π : x → x x −1 = x −1 x, be the projection of S onto its maximal semilattice E = {x ∈ S : x x = x}. Following [3], we define a topological Clifford semigroup S to be a ditopological Clifford semigroup if for any point x ∈ S and a neighborhood O x ⊂ S of x there are neighborhoods U x ⊂ S and W π(x) ⊂ E of x and π(x), respectively, such that (U x ÷ W π(x) ) ∩ π −1 (W π(x) ) ⊂ O x . Ditopological Clifford semigroups are particular cases of ditopological unosemigroups introduced and studied in [3], where the following proposition is proved.

Proposition 2.6 The class of ditopological Clifford semigroups contains all compact topological Clifford semigroups, all topological groups, all topological semilattices, and is closed under taking Clifford subsemigroups, Tychonoff products, and reduced products.
This proposition implies: Corollary 2.7 Let E be a topological semigroup, I be a closed ideal in E, and G be a topological group. Then the reduced product E × I G is a ditopological Clifford semigroup. In particular, for every idempotent e ∈ E the reduced product E × I e G is a ditopological Clifford semigroup. Consequently, the 0-extensionĠ = 2 × {0} G and the cone G = I × {0} G over the topological group G are ditopological Clifford semigroups.

Two embedding theorems
In this section we prove two embedding theorems for ditopological Clifford Usemigroups. For compact topological Clifford semigroups these embedding theorems were proved by O. Hryniv [8]. We recall that a topological Clifford U -semigroup is a topological Clifford semigroup S whose maximal semilattice E is a U -semilattice. The map π : S → E, π : x → x x −1 = x −1 x, will be called the projection of S onto its maximal semilattice E. For every idempotent e ∈ E the preimage π −1 (e) coincides with the maximal subgroup H e of S containing the idempotent e.

The first embedding theorem
Given any idempotent e ∈ E, consider the ideal I e = E \ ⇑e = E \ Int(↑e) in E and the reduced product E × I e H e . Let π e : E × I e H e → E denote the natural projection.
Consider the continuous semigroup homomorphism h e : S → E × I e H e defined by the formula otherwise.
For any subset A ⊂ E the homomorphisms h e , e ∈ A, compose a continuous homomorphism First, we show that h A is an injective homomorphism. Take two distinct points x, y ∈ S and consider their projections π(x), π(y) on the maximal semilattice E.
If π(x) = π(y), then it follows immediately that h a (x) = h a (y) for all a ∈ A, which implies h A (x) = h A (y).
If π(x) = π(y), then the open set U = {e ∈ E : xe = ye} ⊂ E is a neighborhood of the idempotent e = π(x) = π(y). The U -density of the set A in E yields an element a ∈ A ∩ V such that e ∈ ⇑a. For this element a we get h a (x) = (e, xa) = (e, ya) = h a (y), which means that h a separates the points x, y ∈ S. Hence h A is injective. Now we prove that the inverse function h −1 It follows from xe = x ∈ U x and the continuity of the multiplication that there is a neighborhood W e ⊂ W e of the idempotent e such that x W e ⊂ U x .
The Taking the diagonal product ofĥ A A with the natural projection π : S → E, we obtain a continuous homomorphism

Theorem 3.2 If S is a ditopological Clifford U -semigroup, then for any U -dense subset A in E the homomorphism
Proof The definition of the map πĥ A A guarantees that it is a continuous homomorphism. To prove that it is injective take any two distinct points x, y ∈ S and consider their projections π(x), π(y) on the maximal semilattice E.
If π(x) = π(y), then πĥ A π(y), then the open set U = {e ∈ E : xe = ye} ⊂ E is a neighborhood of the idempotent e = π(x) = π(y). The U -density of the set A in E yields an element a ∈ A ∩ V with e ∈ ⇑a, and an element b ∈ A ∩ V ∩ ⇑a with e ∈ ⇑b. It follows Since the topological Clifford semigroup S is dicontinuous, there are a neighborhood U x ⊂ S of x and a neighborhood W e ⊂ E of the idempotent e = π(x) such that (U x ÷ W e ) ∩ π −1 (W e ) ⊂ O x . It follows from xe = x ∈ U x and the continuity of multiplication that there is a neighborhood W e ⊂ W e such that x W e ⊂ U x .
The U -density of A in E yields a point a ∈ A ∩ W e such that e ∈ ⇑a and a point b ∈ A ∩ W e ∩ ⇑a such that e ∈ ⇑b. It follows that W = W e ∩ ⇑b is an open neighborhood of the point e = π(x).
Observe We claim that the neighborhood

Some corollaries of embedding theorem
In this section we shall derive some corollaries of the Embedding Theorem 3.2. In the same way Theorem 3.2 implies: Corollary 4.2 Let S be a topological Clifford semigroup whose idempotent band E is a U 2 -semilattice, and A be a U -dense subset in E. Then the following conditions are equivalent: (1) S embeds into the Tychonoff product of topological semilattices and 0-extensions of topological groups; (2) S embeds into the topological Clifford semigroup E × e∈AḢ ⇑e∩A e ; (3) S is ditopological.
Let us recall that given two topological semigroups X, Y , we say that X is Yembeddable if the canonical homomorphism X → Y Hom(X,Y ) is a topological embedding. It is easy to see that X is Y -embeddable, if and only if X embeds into some power Y κ of Y . In this case we shall write X → Y κ . Corollary 4.3 Let S be a topological Clifford U -semigroup, E be its maximal semilattice, and H = e∈E H e be the Tychonoff product of its maximal subgroups. The following conditions are equivalent: (1) S embeds into a Tychonoff product of the cones over topological groups; (2) S is H -embeddable; (3) S is ditopological and E is I-embeddable.
By Theorem 3.2, S embeds into the Tychonoff product E × e∈E H E e , and the latter product embeds into the Tychonoff product H Hom(E,I) × H E×E , which implies that S is H -embeddable.
In the same way Theorem 3.2 implies: Corollary 4.4 Let S be a topological Clifford U -semigroup, E be its maximal semilattice, and H = e∈E H e be the Tychonoff product of its maximal subgroups. The following conditions are equivalent: (1) S embeds into a Tychonoff product of the 0-extensions of topological groups; (2) S isḢ -embeddable; (3) S is ditopological and E is 2-embeddable.

Characterizing precompact topological Clifford U-semigroups
In this section we shall apply Embedding Theorem 3.2 to prove a compactification theorem for topological Clifford U -semigroups. We recall that a topological semigroup S is called precompact if S embeds into a compact topological semigroup.
It is well-known [1, 3.7.16] that a topological group G is precompact if and only if G is totally bounded, which means that for each non-empty open set U ⊂ G there is a finite subset F ⊂ G such that FU = G = U F. Theorem 3.2 combined with Proposition 2.6 imply the following characterization of precompact topological Clifford U -semigroups.

Theorem 5.1 A topological Clifford U -semigroup S is precompact if and only if the following conditions are satisfied:
(1) the maximal semilattice E = {e ∈ S : ee = e} of S is precompact,  [1, 3.3.14], a topological group G is metrizable by a subinvariant metric if and only if G is first countable and balanced. The latter means that for every neighborhood U ⊂ G of the unique idempotent e of G there is a neighborhood U ⊂ U of e such that x V x −1 = V for all x ∈ G. A simple example of a metrizable topological group, which is not balanced (and hence not metrizable by a subinvariant metric) is the group Aff(R) of affine transformations of the real line.

Theorem 6.1 Each second countable precompact topological Clifford semigroup S is metrizable by a subinvariant metric.
Proof By Proposition 2.3, S embeds into a second countable compact topological Clifford semigroup K , which is metrizable by some metric d. The continuity of the semigroup operation and the inversion on the compact space K implies that the metric is a well-defined subinvariant continuous metric on K . By the compactness of K , this metric generates the topology of K and its restriction ρ|S × S is a subinvariant metric generating the topology of S. Now we turn to the metrization problem for ditopological Clifford U -semigroups. With help of Theorem 3.2 this problem can be reduced to the problem of metrization of semilattices and cones over topological groups.
The latter problem is quite simple. Assume that a topological group H is metrizable by a metric d. Then the cone H over H is metrizable by the metriĉ d(x, y) = min{t x + t y , |t x − t y | + d(h x , h y )}, x, y ∈Ĥ , where (t x , x), (t y , y) ∈ [0, 1] × H are any pairs such that x = q(t x , x) and y = q(t y , y). Here q : [0, 1] × H → H is the canonical projection. If the metric d on H is (left or right) subinvariant, then so is the metricd on H .
This fact combined with Theorem 3.2 implies the following metrization theorem, which generalizes some metrization theorems proved in [2]. Let us remark that by Proposition 2.5, each second countable U -semilattice contains a countable U -dense subset A ⊂ E.

Theorem 6.2 Assume that S is a ditopological Clifford U -semigroup and A is a countable U -dense subset on S. The topological semigroup S is
Theorem 6.2 reduces the problem of (subinvariant) metrizability of a topological Clifford U -semigroup S to the problem of (subinvariant) metrizablity of the maximal subsemilattice E of S. Surprisingly, but this problem is not trivial as witnessed by the following simple example. Example 6.3 There exists a topological semilattice E having the following properties: (1) E is countable and locally compact; (2) E is Lawson; (3) E is metrizable; (4) E is not metrizable by a subinvariant metric; (5) E is not precompact.
Proof Consider a semilattice E = {0}∪{ 1 n } n∈N endowed with the semilattice operation of minimum. Endow E with the topology τ in which all non-zero points are isolated and the sets B n = {0} ∪ { 1 2k } k n , n ∈ N, form a neighborhood base at 0. It is clear that E is a topological semilattice satisfying the conditions (1)-(3) of Example 6.3.
Assume that E is metrizable by a subinvariant metric d. Then for every n ∈ N and the points x = 0, y = 1 2n , z = 1 2n+1 , we get d(0, 1 2n+1 ) = d(zx, zy) ≤ d(x, y) = d(0, 1 2n ) −→ n→∞ 0, which implies that the sequence 1 2n+1 tends to zero. But this contradicts the choice of the topology on E. Therefore, E cannot be metrizable by a subinvariant metric. By Theorem 6.1, the second countable topological semilattice E is not precompact.
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