Absolutely closed semigroups

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Introduction and main results
In many cases, completeness properties of various objects of General Topology or Topological Algebra can be characterized externally as closedness in ambient objects.For example, a metric space X is complete if and only if X is closed in any metric space containing X as a subspace.A uniform space X is complete if and only if X is closed in any uniform space containing X as a uniform subspace.A topological group G is Raȋkov complete if and only if it is closed in any topological group containing G as a subgroup.
On the other hand, for topological semigroups there are no reasonable notions of (inner) completeness.Nonetheless we can define many completeness properties of semigroups via their closedness in ambient topological semigroups.
A topological semigroup is a topological space X endowed with a continuous associative binary operation X × X → X , (x, y) → x y.Definition 1.1 Let C be a class of topological semigroups.A topological semigroup X is called For any topological semigroup we have the implications: Definition 1.2 A semigroup X is defined to be (injectively, absolutely) C-closed if X endowed with the discrete topology has the corresponding closedness property.
We will be interested in the (absolute, injective) C-closedness for the classes: • T 1 S of topological semigroups satisfying the separation axiom T 1 ; • T 2 S of Hausdorff topological semigroups; • T z S of Tychonoff zero-dimensional topological semigroups.
A topological space satisfies the separation axiom T 1 if all its finite subsets are closed.A topological space is zero-dimensional if it has a base of the topology consisting of clopen (= closed-and-open) sets.It is well-known (and easy to see) that every zero-dimensional T 1 topological space is Tychonoff.Since T z S ⊆ T 2 S ⊆ T 1 S, for every semigroup we have the implications: Many examples distinguishing various categorical closedness properties can be found in [9].
A semigroup X is called • chain-finite if any infinite set I ⊆ X contains elements x, y ∈ I such that x y / ∈ {x, y}; • singular if there exists an infinite set A ⊆ X such that A A is a singleton; • periodic if for every x ∈ X there exists n ∈ N such that x n is an idempotent; • bounded if there exists n ∈ N such that for every x ∈ X the n-th power x n is an idempotent; • group-finite if every subgroup of X is finite; • group-bounded if every subgroup of X is bounded; • group-commutative if every subgroup of X is commutative.
The following theorem (proved in [8]) characterizes C-closed commutative semigroups.Theorem 1.4 (Banakh-Bardyla) Let C be a class of topological semigroups such that T z S ⊆ C ⊆ T 1 S. A commutative semigroup X is C-closed if and only if X is chain-finite, nonsingular, periodic, and group-bounded.
The quotient semigroup of X by this congruence is denoted by X /I and called the quotient semigroup of X by the ideal I .If I = ∅, then the quotient semigroup X /∅ can be identified with the semigroup X .
Theorem 1.4 implies that each subsemigroup of a C-closed commutative semigroup is C-closed.On the other hand, quotient semigroups of C-closed commutative semigroups are not necessarily C-closed, see Example 1.8 in [8].This motivates the following notions.
It is easy to see that for every semigroup the following implications hold: Observe that a semigroup X is absolutely C-closed if and only if for any congruence ≈ on X the semigroup X / ≈ is injectively C-closed.
For a semigroup X , let be the set of idempotents of X .
For an idempotent e of a semigroup X , let H e be the maximal subgroup of X that contains e.The union H (X ) = e∈E(X ) H e of all subgroups of X is called the Clifford part of S. A semigroup X is called Ideally and projectively C-closed commutative semigroups were characterized in [8] as follows.
Theorem 1.6 (Banakh-Bardyla) Let C be a class of topological semigroups such that T z S ⊆ C ⊆ T 1 S.For a commutative semigroup X the following conditions are equivalent: (1) X is projectively C-closed; (2) X is ideally C-closed; (3) the semigroup X is chain-finite, group-bounded and Clifford + finite.
In [9] it is shown that the injective (and absolute) T 1 S-closedness is tightly related to the (projective) T 1 S-discreteness.
In Propositions 3.2 and 3.3 of [9] the following two characterizations are proved.
Theorem 1.8 (Banakh-Bardyla) A semigroup X is (1) injectively T 1 S-closed if and only if X is T 1 S-closed and T 1 S-discrete; (2) absolutely T 1 S-closed if and only if X is projectively T 1 S-closed and projectively T 1 Sdiscrete.
The following two theorems characterizing absolutely C-closed commutative semigroups are the main results of this paper.In contrast to Theorems 1.4 and 1.6, the characterizations of absolutely C-closed semigroups essentially depend on the class C, where we distinguish two cases: C = T 1 S and T z S ⊆ C ⊆ T 2 S. Theorem 1.9 For a commutative semigroup X the following conditions are equivalent: (1) X is absolutely T 1 S-closed; (2) X is projectively T 1 S-closed and projectively T 1 S-discrete; (3) X is projectively T z S-closed and projectively T z S-discrete; (4) X is finite.Theorem 1.10 Let C be a class of topological semigroups such that T z S ⊆ C ⊆ T 2 S. For a commutative semigroup X the following conditions are equivalent: (1) X is absolutely C-closed; (2) X is ideally C-closed, injectively C-closed and bounded; (3) X is ideally C-closed, group-finite and bounded; (4) X is chain-finite, bounded, group-finite and Clifford + finite.Theorems 1.9 and 1.10 imply that the absolute C-closedness of commutative semigroups is inherited by subsemigroups: Historical Remark 1.12 Corollary 1.11 does not generalize to noncommutative groups: by Theorem 1.17 in [9], every countable bounded group G without elements of order 2 is a subgroup of an absolutely T 1 S-closed countable simple bounded group X .If the group G has infinite center, then G is not injectively T z S-closed by Theorem 1.13(2) below.On the other hand, G is a subgroup of the absolutely T 1 S-closed group X .This example also shows that the equivalences (1) ⇔ (4) in Theorems 1.9 and 1.10 do not hold for non-commutative groups.
For a semigroup X let be the center of X , and The following theorem proved in [8,Sect. 5] and [10] describes some properties of the center of a semigroup possessing various closedness properties.Theorem 1.13 (Banakh-Bardyla) Let X be a semigroup.
(1) If X is T z S-closed, then the center Z (X ) is chain-finite, periodic and nonsingular.
In [10] it was proved that the (ideal) C-closedness is inherited by the ideal center: Theorem 1.14 (Banakh-Bardyla) Let C be a class of topological semigroups such that T z S ⊆ C ⊆ T 1 S.For any (ideally) C-closed semigroup X , its ideal center IZ (X ) is (ideally) C-closed.Theorem 1.14 suggests the following problem.

Problem 1.15 Let C be a class of topological semigroups. Is the (ideal) center of any absolutely C-closed semigroup X absolutely C-closed?
The "ideal" version of Problem 1.15 has an affirmative answer.Theorem 1. 16 Let C be a class of topological semigroups such that either C = T 1 S or T z S ⊆ C ⊆ T 2 S. Every absolutely C-closed semigroup X has absolutely C-closed ideal center IZ (X ).
The "non-ideal" version of Problem 1.15 has an affirmative answer for group-commutative Z -viable semigroups.
Following Putcha and Weissglass [60] we call semigroup X viable if for any x, y ∈ X with {x y, yx} ⊆ E(X ) we have x y = yx.This notion can be localized using the notion of a viable idempotent.
An idempotent e in a semigroup X is defined to be viable if the set is a coideal in X in the sense that X \ H e e is an ideal in X .By VE(X ) we denote the set of viable idempotents of a semigroup X .
By Theorem 3.2 of [4], a semigroup X is viable if and only if each idempotent of X is viable if and only if for every x, y ∈ X with x y = e ∈ E(X ) we have xe = ex and ye = ey.This characterization implies that every semigroup X with E(X ) ⊆ Z (X ) is viable.In particular, every commutative semigroup is viable.
For ideally (absolutely) T z S-closed semigroups we have the following description of the structure of maximal subgroups of viable idempotents, see [10,Theorem 1.7].
Theorem 1.17 (Banakh-Bardyla) Let e be a viable idempotent of a semigroup X and H e be the maximal subgroup of e in X .
(1) If X is ideally T z S-closed, then the group Z (H e ) is bounded.
(2) If X is absolutely T z S-closed, then the group Z (H e ) is finite.
, if each central idempotent of X is viable.It is clear that each viable semigroup is Z -viable.On the other hand, there exist semigroups which are not Z -viable, see Remark 2.6.
For a subset A of semigroup X let In fact, the "ideal" part of Theorem 1.14 was derived in [10] from the following theorem, which will be essentially used also in this paper: The following theorem gives a partial answer to Problem 1.15 for the class C = T 1 S.
The cardinal cov(M) appearing in Theorem 1.20(3) is defined as the smallest cardinality of a cover of the real line by nowhere dense subsets.The Baire Theorem implies that ω 1 ≤ cov(M) ≤ c.It is well-known that cov(M) = c under Martin's Axiom.By [17, 7.13], the equality cov(M) = c is equivalent to Martin's Axiom for countable posets.
By Theorem 1.13(1), the center Z (X ) of any T z S-closed semigroup is chain-finite.In fact, this is an order property of the poset E(X ) endowed with the natural partial order ≤ defined by x ≤ y iff x y = yx = x.In turns out that stronger closedness properties (like the ideal or projective C-closedness) impose stronger restrictions on the partial order of the set E(X ) and also on the partial order of the semilattice reflection X / of X .
A congruence ≈ on a semigroup X is called a semilattice congruence if the quotient semigroup X / ≈ is a semilattice, i.e., a commutative semigroup of idempotents.The intersection of all semilattice congruences on a semigroup X is called the smallest semilattice congruence on X and the quotient semigroup X / is called the semilattice reflection of X .The smallest semilattice congruence was studied in the monographs [18,53], surveys [54,55] and papers [3,[57][58][59][60][65][66][67].
A partially ordered set (P, ≤) is called • chain-finite if each infinite subset I ⊆ P contains two elements x, y such that x ≤ y and y ≤ x; • well-founded if each nonempty set A ⊆ P contains an element a such that {x ∈ A : x ≤ a} = {a}.It is easy to see that for every chain-finite semigroup X the poset E(X ) is chain-finite.The converse holds if E(X ) is a commutative subsemigroup of X .Theorem 1.21 Let X be a semigroup.
(1) If X is ideally T z S-closed, then the posets X / and VE(x) are well-founded.
(2) If X is projectively T z S-closed, then X / and VE(x) are chain-finite.
(3) If X is projectively T z S-closed and projectively T z S-discrete, then X / and VE(x) are finite; (4) If X is absolutely T 1 S-closed, then X / and VE(x) are finite.
Theorem 1.21 will be proved in Sect.3. In Sect. 4 we prove a general version of Theorem 1.9 and in Sect. 5 we prove Lemma 5.2 giving a sufficient condition of the absolute T 2 S-closedness.In Sect.6 we introduce the notion of an A-centrobounded semigroup and use this notion for characterizing bounded set of form Z (X )∩ N √ A in absolutely T z S-closed semigroups.In Sect.7 we prove Theorem 7.1 giving some sufficient conditions of the A-centroboundedness and implying Corollary 7.2, which is a more general version of Theorem 1.20.In Sects.8 and 9 we prove Theorems 1.16 and 1.10, respectively.
We denote by ω the set of finite ordinals and by N def = ω\{0} the set of positive integers.Each ordinal n ∈ ω is identified with the set {k : k < n} of smaller ordinals.

Partially ordered sets
A poset is a set endowed with a partial order ≤.For an element p of a poset P, let ↓ p def = {x ∈ P : x ≤ p} and ↑ p def = {x ∈ P : p ≤ x} be the lower and upper sets of p in P, respectively.

Let
• cov(M) be the smallest cardinality of a cover of the real line by nowhere dense sets, • cov(N) be the smallest cardinality of a cover of the real line by sets of Lebesgue measure zero, • cov(N) be the smallest cardinality of a cover of the real line by closed subsets of Lebesgue measure zero, and • d be the smallest cardinality of a cover of the Baire space ω ω by compact sets.

Semigroup topologies and subinvariant metrics on semigroups
A topology τ on a semigroup X is called a semigroup topology if (X , τ ) is a topological semigroup.
A metric d on a semigroup X is subinvariant if for every x, y, a ∈ X and we have

d(ax, ay) ≤ d(x, y) and d(xa, ya) ≤ d(x, y).
It is easy to see that every subinvariant metric on a semigroup generates a semigroup topology.

Zero-closed semigroups
For a semigroup X its Following [9], we call a semigroup X zero-closed if X is closed in its 0-extension X 0 = {0} ∪ X endowed with any Hausdorff semigroup topology.
A topological semigroup X is called 0-discrete if X contains a unique non-isolated point 0 such that x0 = 0 = 0x for all x ∈ X .It is easy to see that every 0-discrete T 1 topological semigroup is zero-dimensional.

Lemma 2.1 Let C be a class of topological semigroups containing all
Proof Assuming that X is not zero-closed, we can find a Hausdorff semigroup topology τ on X 0 such that X is not closed in the topological space (X 0 , τ ).Consider the topology τ 0 on X 0 , generated by the base {x} : x ∈ X ∪ τ , and observe that (X 0 , τ 0 ) is a 0-discrete Hausdorff topological semigroup containing X as a non-closed subsemigroup and proving that X is not C-closed.

Polybounded and polyfinite semigroups
for some n ∈ N and some elements a 0 , . . ., a n ∈ X 1 .The number n is called the degree of the polynomial f and is denoted by deg Polybounded semigroups were introduced in [9], where it was proved that countable zeroclosed semigroups are polybounded and polybounded groups are absolutely T 1 S-closed.
A semigroup X is called polyfinite if there exist d ∈ N and a finite set F ⊆ X such that for any x, y ∈ X there exists a semigroup polynomial f : Given any elements x, y ∈ X , find i ∈ n such that f i (x) = b i and then find j ∈ n such that The following theorem was proved in [12].
(2) If cov(M) = c and X admits a subinvariant separable complete metric, then X is polybounded.
(3) If cov(M) = c and X admits a compact Hausdorff semigroup topology, then X is polybounded.(4) If cov(N) = c and X admits a compact Hausdorff semigroup topology, then X is polyfinite.

Prime coideals in semigroups
is a homomorphism from X to the semilattice {0, 1} endowed with the operation of minimum.

Lemma 2.4 If a semigroup X is absolutely (resp. projectively) T z S-closed, then any prime coideal in X is absolutely (resp. projectively) T z S-closed.
Proof Assume that a semigroup X is absolutely (resp.projectively) T z S-closed and let C be a prime coideal in X .To prove that the semigroup C is absolutely (resp.projectively) By the absolute (resp.projective) T z S-closedness of X , the image h[X ] is closed in (Y 0 , τ 0 ) and then the set ), proving that the semigroup C is absolutely (resp.projectively) T z S-closed.

Viable idempotents in semigroups
We recall that an idempotent e of a semigroup X is viable if the subsemigroup H e e def = {x ∈ X : xe = ex ∈ H e } is a prime coideal in X .By VE(X ) we denote the set of viable idempotents in X .

Lemma 2.5 For any semigroup X we have E(IZ(X
Historical Remark 2. 6 The inclusion E(Z ) ∩ IZ(X ) ⊆ VE(X ) in Lemma 2.5 cannot be improved to the inclusion E(X ) ∩ Z (X ) ⊆ VE(X ): by [14] or [20] there exist infinite congruence-free monoids.In every congruence-free monoid X = {1} the idempotent 1 is central but not viable.

Proof of Theorem 1.21
In this section, for any semigroup X we study the order properties of the posets VE(X ) and X / and prove Theorem 1.21.By 2 we denote the two-element semilattice {0, 1} endowed with the operation of minimum.
Proposition 3.1 Let X be a semigroup and q : X → X / be the quotient homomorphism onto its semilattice reflection.The restriction q VE(X ) is injective and hence is an isomorphic embedding of the poset VE(X ) into the poset X / .
Proof Given two viable idempotents e 1 , e 2 ∈ VE(X ), assume that q(e 1 ) = q(e 2 ).For every i ∈ {1, 2}, the definition of a viable idempotent ensures that the semigroup e i , 0, otherwise, is a homomorphism.The equality q(e 1 ) = q(e 2 ) implies that Thus, e 1 e 2 = e 2 e 1 ∈ H e 1 ∩ H e 2 , which implies e 1 = e 2 , and proves that the restriction q VE(X ) is injective.
In the following four lemmas we prove the statements of Theorem 1.21.

Lemma 3.2
For any ideally T z S-closed semigroup X , the posets X / and VE(X ) are wellfounded.
Proof By Proposition 3.1, the poset VE(X ) embeds into the semilattice reflection X / of X , so it suffices to prove that the poset Y def = X / is well-founded.Assuming that Y is not well-founded, we can find a strictly decreasing sequence (y n ) n∈ω in Y .For every n ∈ ω consider the upper set ↑y n = {y ∈ Y : y n ≤ y} and observe that ↑y n is a prime coideal in Y .Consequently, its preimage P n = q −1 (↑y n ) is a prime coideal in X .
It is easy to see that P = n∈ω P n is a subsemigroup of X and the complement I def = X \ P is an ideal in X .Consider the semigroup S = P ∪ {P} ∪ {I } endowed with the semigroup operation * : S × S → S defined by Endow the semigroup S with the topology τ generated by the base and observe that (S, τ ) is a Hausdorff zero-dimensional topological semigroup with a unique non-isolated point P. Since (S, τ ) contains the quotient semigroup X /I = P ∪ {I } as a discrete subsemigroup, the semigroup X is not ideally T zS -closed, which contradicts our assumption.

Lemma 3.3
For any projectively T zS -closed semigroup X , the posets X / and VE(X ) are chain-finite.
Proof Let q : X → X / be the quotient homomorphism of X onto its semilattice reflection.If the semigroup X is projectively T zS -closed, then its semilattice reflection X / is projectively T zS -closed and hence T zS -closed.By Theorem 1.4, the semilattice X / is chain-finite.Then X / is also chain-finite as a poset.By Proposition 3.1, the poset VE(X ) is chain-finite, being order isomorphic to a subset of the chain-finite poset X / .Lemma 3.4 If a semigroup X is projectively T zS -closed and projectively T zS -discrete, then the sets X / and VE(X ) are finite.
Proof Let q : X → X / be the quotient homomorphism of X onto its semilattice reflection.Consider the set H of all homomorphisms from X / to the two-element semilattice 2. Since homomorphisms to 2 separate points of semilattices, the homomorphism δ : X / → 2 H , δ : x → (h(x)) h∈H , is injective.Since the semilattice X / is T zS -closed and T zS -discrete, the image δ[X / ] is a closed discrete subsemilattice of the compact topological semilattice 2 H . Hence X / is finite and so is the set VE(X ).
Theorem 1.8 and Lemma 3.4 imply the following lemma.Lemma 3.5 For any absolutely T 1 S-closed semigroup X , the sets X / and VE(X ) are finite.

Absolutely T 1 S-closed semigroups
In this section we establish some properties of absolutely T 1 S-closed semigroups and prove the following theorem that implies the characterization Theorem 1.9 announced in the introduction.
To prove that (4) ⇒ (5), assume that the semigroup X is projectively T zS -closed and projectively T zS -discrete.By Lemma 3.4 the set VE(X ) is finite and by Theorem 1.13(1,2), the semigroup Z (X ) is periodic and group-finite.Then for every e ∈ E(X ) the intersection Z (X ) ∩ H e is either empty or a finite subgroup of Z (X ).In both cases, the set Z (X ) ∩ H e is finite.Then the set is finite, being the union of finitely many finite sets.
To prove that (5) ⇒ (6), assume that the semigroup X is projectively T zS -closed and the set Z (X ) ∩ H (X ) ∩ N √ VE(X ) is finite.By Theorem 1.13(1), the semigroup Z (X ) is periodic.By Theorem 1.18, the set Z (X ) ∩ ∞ √ VE(X ) \ H (X ) is finite and then the set is finite, too.Now assuming that X is commutative, we shall prove that (6) ⇒ (1).So, assume that the semigroup Z (X ) is periodic and and hence the commutative semigroup X = Z (X ) is finite.

Corollary 4.2 If a Z -viable semigroup is absolutely T 1 S-closed, then its center Z (X ) is finite and hence T 1 S-closed.
Proof The Z -viability of X yields E(X ) ∩ Z (X ) ⊆ VE(X ).By Theorem 1.13(1), the semigroup Z (X ) is periodic and hence

A sufficient condition of the absolute T 2 S-closedness
In this section we shall prove a sufficient condition of the absolute T 2 S-closedness.We shall use the following theorem, proved by Stepp in [63, Theorem 9].

Theorem 5.1 (Stepp) Every chain-finite semilattice is absolutely T 2 S-closed.
A semigroup X is called E-commutative if x y = yx for any idempotents x, y ∈ E(X ).

Lemma 5.2 Each chain-finite group-finite bounded Clifford + finite E-commutative semigroup X is absolutely T 2 S-closed.
Proof To show that X is absolutely T 2 S-closed, take any homomorphism h : X → Y to a Hausdorff topological semigroup Y .We should prove that the semigroup h[X ] is closed in Y .Replacing Y by h[X ], we can assume that h[X ] is dense in Y .Since X is bounded, there exists n ∈ N such that x n ∈ E(X ) and hence x 2n = x n for every x ∈ X .Taking into account that h is a homomorphism, we conclude that y 2n = y n for all y ∈ h[X ].The closed subset {y ∈ Y : y 2n = y n } of Y contains the dense set h[X ] and hence coincides with Y .Therefore, y n ∈ E(Y ) for all y ∈ Y .It follows that the continuous map φ : Y → E(Y ), φ : y → y n , is well-defined.Consider the function ψ : X → E(X ), ψ : x → x n , and observe that h Since X is a chain-finite E-commutative semigroup, the set E(X ) is a chain-finite subsemilattice of X .By Theorem 5.1, the chain-finite semilattice E(X ) is absolutely T 2 S-closed and hence its image h[E(X )] is closed in the Hausdorff topological semigroup Y .The continuity of the map φ : Since the semilattice E(X ) is chain-finite, we can apply Theorem 1.6 and conclude that the semilattice h[E(X )] = E(Y ) is chain-finite and so is the subsemilattice and the semilattice E(X ) is chain-finite, the nonempty subsemilattice h −1 [↑e] ∩ E(X ) has a unique minimal element μ ∈ h −1 (e).Since the semigroup X is group-finite, the maximal subgroup H μ is finite and so is the set h y .Since φ(ey) = (ey) n = (y n+1 ) n = (y n ) n+1 = e n+1 = e ∈ ↑e, we can additionally assume that φ[eO y ] ⊆ ↑e. Since we can choose an element x ∈ H (X )∩h −1 [O y ] and observe that h((μx

Bounded sets in absolutely T zS -closed semigroups
In this section, given an absolutely T zS -closed semigroup X , we characterize subsets A ⊆ VE(X ) for which the set Z (X ) The following notion plays a crucial role in our subsequent results.Definition 6.1 A semigroup X is defined to be A-centrobounded over a set A ⊆ E(X ) if there exists n ∈ N such that for every e ∈ A and x, y ∈ a∈A H a a with (xe)(ye) −1 ∈ H e ∩ Z (X ) we have ((xe)(ye) −1 ) n ∈ E(X ).
In the following theorem we endow the set VE(X ) with the natural partial order ≤ considered in Sect.3. A subset A ⊆ VE(X ) is called an antichain if x ≤ y for any distinct elements x, y ∈ A. Theorem 6.2 Let X be an absolutely T zS -closed semigroup.For a subset A ⊆ VE(X ) the following conditions are equivalent:

) X is B-centrobounded over every countable infinite antichain B ⊆ A.
Proof Replacing the semigroup X by its 1-extension X 1 , we lose no generality assuming that the semigroup X contains a two-sided unit 1.By Theorem 1.13(1,2), the semigroup Z (X ) is chain-finite, periodic, nonsingular, and group-finite.
The equivalence (1) ⇔ (2) follows from Theorem 1.18 and (2) ⇒ (3) is trivial.Indeed, by (2), there exists We claim that the number n witnesses that X is B-centrobounded over any set B ⊆ A. Indeed, given any idempotent e ∈ B and elements x, y ∈ b∈B H b b with (xe)(ye) −1 ∈ Z (X ), by the periodicity of Z (X ) and the choice of n, we have and hence ((xe)(ye) −1 ) n ∈ E(X ).

It remains to prove the implication (3) ⇒ (2).
Let π : N √ E(X ) → E(X ) be the map assigning to each element x ∈ N √ E(X ) a unique idempotent π(x) in the monogenic semigroup x N def = {x n : n ∈ N}.To derive a contradiction, assume that the condition (3) is satisfied but (2) is not.Claim 6.3There exists a sequence Let [ω] 2 be the family of two-element subsets of ω.Consider the function χ : [ω] 2 → {0, 1, 2} defined by By the Ramsey Theorem (see [41,Theorem 5]), there exists an infinite set ⊆ ω such that χ[[ ] 2 ] = {c} for some c ∈ {0, 1, 2}.If c = 0, then the set {π(g n )} n∈ contains a unique idempotent u and hence the set {g k } k∈ ⊆ Z (X ) ∩ H u is finite (since Z (X ) is periodic and group-finite).By the Pigeonhole Principle, for any k > |Z (X ) ∩ H u | there are two numbers i < j ≤ k such that g i k = g j k , which contradicts the choice of g k .Therefore, c = 0.If c = 1, then the set {π(g k )} k∈ is an infinite chain in E(X ) ∩ Z (X ) which is not possible as Z (X ) is chain-finite.Therefore, c = 2 and hence {π(g k )} k∈ is an infinite antichain in E(Z (X )).Write the infinite set as {n k } k∈ω for some strictly increasing sequence (n k ) k∈ω .For every k ∈ ω put z k = g n k and observe that the sequence (z k ) k∈ω satisfies the conditions (1), (2) of Claim 6.3.
Let q : X → X / be the quotient homomorphism of X onto its semilattice reflection.Let (z n ) n∈ω be the sequence from Claim 6.3.For every n ∈ ω let e n = π(z n ).The inclusion z n ∈ Z (X ) ∩ N √ A and the periodicity of Z (X ) imply that the idempotent is a prime coideal in X and moreover, H en e n = q −1 [↑q(e n )], see Proposition 2.15 in [4].Since the semigroup X is absolutely T zS -closed, for the ideal For convenience, by 0 we denote the element I ∈ X /I .The injectivity of the restriction q VE(X ) implies that e n e m ∈ I for any distinct n, m ∈ ω.This implies that the ideal I is not empty and the element 0 = I of the semigroup X /I is well-defined.Now we introduce a 0-discrete Hausdorff semigroup topology τ on the semigroup Fix any free ultrafilter F on N. Let Note that the set Q is nonempty, as q(1) ∈ Q (we assumed that X contains a unit exactly to omit the easier case Q = ∅).
For any y 1 , y 2 ∈ Q there exist F 1 , F 2 ∈ F such that q(e n ) ≤ y i for each n ∈ F i , i ∈ {1, 2}.Then q(e n ) ≤ y 1 y 2 for each n ∈ F 1 ∩ F 2 ∈ F. It follows that Q is a subsemilattice of X / .By Lemma 3.3, the semilattice X / is chain-finite and so is its subsemilattice Q.Thus, the semilattice Q contains the smallest element s.Since s ∈ Q, there exists a set F s ∈ F such that q(e n ) ≤ s for all n ∈ F s .Consider the prime coideal Claim 6.4 ↑s = n∈F s ↑q(e n ) and hence q −1 (↑s) = C.

Proof
The inclusion ↑s ⊆ n∈F s ↑q(e n ) follows from the choice of F s .Now take any y ∈ n∈F s ↑q(e n ) and observe that {n ∈ ω : q(e n ) ≤ y} ∈ F and hence y ∈ Q and s ≤ y by the choice of s.Then To introduce the topology τ on Y , we need the following notations.For a real number r by r def = max{n ∈ Z : n ≤ r } we denote the integer part of r .For each k ∈ N and n ∈ F s let

The definition of the set A(n, k) implies that
of Y .On the semigroup Y = X /I consider the topology τ generated by the base

Claim 6.5 (Y , τ ) is a Hausdorff zero-dimensional topological semigroup.
Proof To see that the topology τ is Hausdorff, it suffices to show that for any y ∈ Y \ {0} there exist k ∈ N and Therefore, the topology τ is Hausdorff.Since 0 is a unique non-isolated point of (Y , τ ), the topology τ is zero-dimensional.
It remains to prove that (Y , τ ) is a topological semigroup.Given any two points y, y ∈ Y and a neighborhood O yy ∈ τ of their product yy ∈ Y , we need to find neighborhoods So, it remains to consider three cases: (1) y = 0 and y = 0; (2) y = 0 and y = 0; (3) y = 0 = y .
In each of these cases, yy = 0, so we can find F ∈ F and k ∈ N such that F ⊆ F s and U k,F ⊆ O yy .
(1) Assume that y = 0 and y = 0.If y ∈ C, then So, we can put O y = {y} and O y = U k,F .If y / ∈ C, then q(y) / ∈ Q and hence the set {n ∈ N : q(e n ) ≤ q(y)} does not belong to the ultrafilter F and then the set G def = {n ∈ F s : q(e n ) ≤ q(y)} belongs to the ultrafilter F. For every n ∈ G and p ∈ N we have q(yz p n ) = q(y)q(e n ) = q(e n ), implying yz (2) The case y = 0 and y = 0 can be treated by analogy with the preceding case.
Note that (Y , τ ), being a continuous homomorphic image of the absolutely T zS -closed semigroup X , is itself T zS -closed.But, as we will show later, this is not the case.
Let z F be the ultrafilter on Y generated by the base {z F : F ∈ F} where z F def = {z n : n ∈ F} for F ∈ F. Note that for any y ∈ Y the filter yz F generated by the base {yG : G ∈ z F } is an ultrafilter on Y .Also, since {z n : n ∈ N} ⊆ Z (X ) we get that yz F = z F y, where z F y is the ultrafilter generated by the base {Gy : G ∈ z F }. Claim 6. 6 For any y ∈ Y \ C the ultrafilter yz F is the principal ultrafilter at 0. Proof The claim is obvious if y = 0. So, assume that y ∈ Y \{0} = X \I .Since y ∈ Y \C, the set {n ∈ N : q(e n ) ≤ q(y)} does not belong to the ultrafilter F. Then the set F = {n ∈ F s : q(e n ) ≤ q(y)} belongs to F. Now observe that yz F ⊆ I and hence yz F = {0}.Therefore, the ultrafilter yz F is principal at 0. Claim 6.7 There exists m ∈ N such that U m,F s / ∈ yx F for every y ∈ C.
Proof Since X is {e n } n∈F s -centrobounded, there exists m ∈ N such that for every n ∈ F s and We claim that for every y ∈ C, the set U m,F s does not belong to the ultrafilter yz F .In the opposite case, the set U m,F s has non-empty intersection with the set yz F s .Then there exists n ∈ F s such that yz n ∈ U m,F s and hence z n y = yz n = z which contradicts the choice of the point z n in Claim 6.
Let T = Y ∪ {yz F : y ∈ C}.Extend the semigroup operation from Y to the set T by the formula: Let θ be the topology on the semigroup T which satisfies the following conditions:

Claim 6.8 The topology θ on T is Hausdorff and zero-dimensional.
Proof First we show that the topological space (T , θ) is zero-dimensional.Given an open set U ∈ θ and a point u ∈ U , we need to find a clopen set V in (T , θ) such that u ∈ V ⊆ U .We consider three possible cases.
1.If u ∈ Y \ {0}, then u is an isolated point of Y and T .So, we can take V = {u}.The definition of the topology θ ensures that V = {u} is a clopen neighborhood of u in (T , θ).
2. If u = 0, then we can apply Claim 6.7 and find m ∈ N and F ∈ F such that U m,F ⊆ U and U m,F / ∈ yz F for all y ∈ C. The definition of the topology θ ensures that V = U m,F is a clopen neighborhood of u = 0 in (T , θ).
3. If u = yz F for some y ∈ C, then by the definition of the topology θ , there exists F ∈ F such that F ⊆ F s and yx F ⊆ U .Moreover, by Claim 6.7, we can assume that yx F ∩ U m,F s = ∅ for some m ∈ N. By the definition of the topology θ , the set Therefore the topology θ is zero-dimensional and being T 1 , it is Hausdorff.
To check the continuity of the semigroup operation in (T , θ), take any elements a, b ∈ T and choose any neighborhood O ab ∈ θ of their product ab.We must find neighborhoods 2. a = yz F for some y ∈ C and b ∈ Y .This case can be considered by analogy with the preceding case.
3. a = yz F and b = y z F for some y, y ∈ C. In this case ab = 0 and we can find Observe that the continuity of the binary operation in (T , θ) implies that it is associative, as Y is a dense subsemigroup of T .Thus, (T , θ) is a Hausdorff zero-dimensional topological semigroup which contains (Y , τ ) as a non-closed subsemigroup.But this contradicts the absolute T zS -closedness of (Y , τ ) and X .The obtained contradiction completes the proof of the implication (3) ⇒ (2) and also the proof of Theorem 6.2.

Some sufficient conditions of centroboundedness
In this section we shall find some sufficient conditions for centroboundedness, which will be combined with Theorem 6.2 in order to obtain the boundedness of certain sets in absolutely T zS -closed semigroups.
The following theorem is the main result of this section.
2. Assume that the semigroup h[C] is polyfinite.Then there exist n ∈ N and a finite set F ⊆ h[C] such that for any x, y ∈ h[C] there exists a semigroup polynomial f : h For every k ∈ N the assumption z = (xe)(ye) −1 ∈ H e ∩Z (X ) implies xe = zye and hence (xe) k = (zye) k = z k (ye) k .By the choice of n and F, there exists a semigroup polynomial  It follows from x i e = z i y i e and z ∈ Z (X ) that a = fi (x i e) = fi (z i y i e) = z i deg ( fi )  fi (y i e) = z id b.
Similarly we can prove that a = z jd b.Since H e is a group, the equality z id b = z jd b implies z id = z jd and z  To derive a contradiction, assume that the semigroup X is not A-centrobounded.Writing down the negation of the A-centroboundedness, we obtain sequences (x + n ) n∈ω , (x − n ) n∈ω , (z n ) n∈ω and (e n ) n∈ω such that for every n ∈ ω the following conditions are satisfied:  Recall that μ n is the least common multiple of the numbers exp(Ce k ), k < n, and observe that for every x ∈ h [C] ⊆ n∈ω Ce n , its inverse x −1 in H is the limit of the sequence (x 1+μ n ) n∈ω , which implies that x −1 ∈ h [C] = h [C] and means that h [C] is a subgroup of H .
For every n ∈ ω consider the element Observe that for every k < n we have

Corollary 1 .
11 Let C be a class of topological semigroups such that either C = T 1 S or T z S ⊆ C ⊆ T 2 S. Every subsemigroup of an absolutely C-closed commutative semigroup is absolutely C-closed.
O y , O y ∈ τ of y, y , respectively, such that O y O y ⊆ O yy .If y, y ∈ Y \{0}, then the neighborhoods O y = {y} and O y = {y } have the required property: O y O y = {yy } ⊆ O yy .
p n ∈ I and yU 1,G = {0}.So we can put O y = {y} and O y = U 1,G .
If y = 0 = y , then we can put O y = O y = U 4k,F .Let us show that O y O y ⊆ U k,F .Indeed, take any a, b ∈ U 4k,F .If 0 ∈ {a, b} or a and b do not belong to the same subgroup H e n , then ab = 0 ∈ U k,F .Otherwise, there exists n ∈ N such that a, b ∈ A(n, 4k) and hence a ∈ z m n C and b

p n c
for some c ∈ C and p ∈ N with 2 ≤ p ≤ 2 n /m .It follows from c, y ∈ C ⊆ H en e n that ce n , ye n ∈ H e n .Then the equality z n y = z p n c implies that z n ye n = z p n ce n ∈ H e n and hence which contradicts the choice of t / ∈ V .Therefore, E / ∈ F and the set G def = F\E = {n ∈ F : yz n = cz n } belongs to the ultrafilter F.
O a , O b ∈ θ of a, b such that O a O b ⊆ O ab .If a, b ∈ Y , then such neighborhoods exist by the continuity of the semigroup operation in the topological semigroup (Y , τ ).
So, it remains to consider three cases: 1. a ∈ Y and b = yz F for some y ∈ C.This case has three subcases.(1a) a ∈ C. In this case there exists a set F ∈ F such that ayx F ⊆ O ab , and then the neighborhoods O a = {a} andO b = yx F ∪ {yx F } have the required property O a O b = ayx F ∪ {ayx F } ⊆ O ab .(1b) a ∈ Y \ ({0} ∪ C) = X \ (I ∪ C).Since a / ∈ C, the set {n ∈ ω : q(e n ) ≤ q(a)}does not belong to the ultrafilter F and hence the setF def = {n ∈ F s : q(e n ) ≤ q(a)} belongs to F. Then for the neighborhoods O a = {a} and O b = yz F ∪ {yz F } we have O a O b = {0} = {ab} ⊆ O ab .(1c) a = 0.In this case ab = 0 and we can find k ∈ N and F ∈ F such that U k,F ⊆ O ab .We claim that the neighborhoods O a = U 2k,F and O b = yz F ∪ {yz F } satisfy O a O b ⊆ O ab .Given any elements a ∈ O a and b ∈ O b , we should check that a b ∈ O ab .This is clear if a = 0.If a = 0, then a = z p n y for some n ∈ F, 2 ≤ p ≤ 2 n /2k and y ∈ C. If b = yz F , then a b = 0 ∈ O ab .So, assume that b = yz n for some n ∈ F. If n = n , then a b = z p n y yz n ∈ I and hence a b = 0 = ab ∈ O ab .Suppose that n = n .Then a b = z p n y yz n = z p+1 n y y and 2 it suffices to check that for any e ∈ A and x, y ∈ C with z def = (xe)(ye) −1 ∈ Z (X ) we have z m = e where m = (|F|n) 2 !.

For a finite
subset F ⊆ N let lcm(F) def = min{n ∈ N : ∀x ∈ F ∃k ∈ N (n = xk)} be the least common multiple of numbers in the set F.

Corollary 7 . 2
) μ n = ((x + n e k )(x − n e k ) −1 ) μ n = ((x + n e k )(x − n e k ) −1 ) exp(Ce k ) μ n / exp(Ce k ) = e μ n /exp(Ce k ) k = e k , which means that the sequence (x ± n ) n∈ω converges to the identity element e = (e k ) k∈ω of the topological group H = k∈ω H e k .Let 2 = {0, 1} and 2 <ω def = n∈ω 2 n .Define the family ( p s ) s∈2 <ω of elements of H by the recursive formula: p ∅ = e and p sˆ0 = p s , p sˆ1 = p s • x ± n for every n ∈ ω and s ∈ 2 n .The definition of the ultrametric ρ on H and the convergence of x ± n → e imply that for every s ∈ 2 ω the sequence ( p s n ) n∈ω is Cauchy in the metric space (H , ρ ) and hence it converges to some element p s∈ h [C] ⊆ H . Since { p s : s ∈ 2 ω } ⊆ h [C] ⊆ α∈κ f −1 α (b α ) and κ < c, there exists α ∈ κ such that the set 2 ω α def = {s ∈ 2 ω : p s ∈ f −1 α (b α )} is uncountable and hence contains two distinct sequences s, s ∈ 2 ω α such that s deg( f α ) = s deg( f α ) .Let m ∈ N be the smallest number such that s(m) = s (m).Then m ≥ deg( f α ) ≥ 1 and s m = s mby the minimality of m.Let t = s m = s m and observe that { p s (m+1) , p s (m+1) } = { p tˆ0 , p tˆ1 }.We lose no generality assuming that p s (m+1) = p tˆ0 = p t and p s (m+1) = p tˆ1 = p t x ± m .It follows from f α ( p s ) = b α = f α ( p s ) that pr m ( f α ( p s )) = pr m ( f α ( p s )).Find elements a 0 , a 1 , . . ., a deg( f α ) ∈ h [C] such that f α (x) = a 0 xa 1 x • • • xa deg( f α ) for all x ∈ h [C].For every i ∈ {0, . . ., deg( f α )} let ǎi = pr m (a i ).Let fα : H e m → H e m be the semigroup polynomial defined by fα(x) = ǎ0 x ǎ1 x • • • x ǎdeg( f α ) for x ∈ H e m .It is clear that pr m • f α = fα • pr m .It follows from pr m (x ± m ) = z μ m m ∈ Z (X ) and pr m (x ± k ) = e m for all m < k that fα (pr m ( p t )) = fα (pr m ( p s )) = pr m ( f α ( p s )) = pr m ( f α ( p s )) = fα (pr m ( p s )) = fα (pr m ( p tˆ1 )) = fα (pr m ( p t )pr m (x ± m )) = fα (pr m ( p t )z μm m ) = fα (pr m ( p t ))z μm deg fα mand hence e m = zμ m deg f α m, which contradicts the choice of z m .Let X be an absolutely T zS -closed semigroup and A ⊆ VE(X ).Assume that for any infinite countable antichain B ⊆ A, the coideal C def= e∈B H e e and the homomorphism h : C → H def = b∈B H b , h : x → (xe) e∈B , one of the following conditions is satisfied: (1) for every e ∈ B the subsemigroup Ce of H e is commutative; (2) the semigroup h[C] is polyfinite; (3) h[C] is countable; (4) |h[C]| ≤ cov(M), and for every e ∈ A the subsemigroup Ce ⊆ H e is countable.(5) |h[C]| ≤ c and for every e ∈ A the subsemigroup Ce of H e is bounded.The implication (2) ⇒ (3), (3) ⇒ (4), and (4) ⇒ (1) follow from Theorems 1.13(2), 1.6, and Lemma 5.2, respectively.Funding Open access funding provided by The Ministry of Education, Science, Research and Sport of the Slovak Republic in cooperation with Centre for Scientific and Technical Information of the Slovak Republic For classes C with T z S ⊆ C ⊆ T 2 S a partial answer to Problem 1.15 looks as follows.|C| ≤ cov(M), and for every e ∈ A the subsemigroup Ce of H e is countable.(4) |C| ≤ c and for every e ∈ A the subsemigroup Ce of H e is bounded.
X is projectively T zS -closed and h[C] is countable; (4) X is absolutely T zS -closed, |h[C]| ≤ cov(M), and for every e ∈ A the subsemigroupCe ⊆ H e is countable.(5)X is absolutely T zS -closed, |h[C]| ≤ c and for every e ∈ A the subsemigroup Ce of H e is bounded.Proof By our assumption, the set A is countable and hence admits an injective function λ : A → ω.Consider the group H def = e∈A H e endowed with the Tychonoff product topology, where the groups H e , e ∈ A are discrete.This topology is generated by the complete invariant metric ρ : H × H → R defined by ρ((x a ) a∈A , (y a ) a∈A ) = max {0} ∪ 1 2 λ(a) : a ∈ A, x a = y a .For every a ∈ A, let pr a : H → H a pr a : (x e ) e∈A → x a , be the ath coordinate projection.The definition of the homomorphism h implies that pr a • h = h a for every a ∈ A. 1. Assume that X is projectively T zS -closed and for every e ∈ A the subsemigroup Ce of H e is commutative.Then the semigroup h[C] ⊆ e∈A Ce is commutative.By Lemma 2.4, the prime coideal C of X is projectively T zS -closed and so is its homomorphic image h[C].By Theorem 1.13(1), the T zS -closed commutative semigroup h[C] ⊆ H is periodic and hence is a subgroup of the group H .By Theorem 1.4, the T zS -closed commutative group h[C] is bounded.Then there exists n ∈ N such that z n ∈ E(H ) for every z ∈ h[C].Consequently, for every x, y ∈ C and e ∈ A we have that ((xe) e : C → H e , h : x → xe = ex.The homomorphisms h e create the homomorphismh : C → H , h : x → (h e (x)) e∈A = (xe) e∈A .
consider the element ǎi = pr e (a i ) of the group H e .Let fk : H e → H e be the semigroup polynomial defined by fk Assume that the semigroup X is projectively T zS -closed and the set h[C] is countable.By Lemma 2.4, the prime coideal C in X is projectively T zS -closed and so is its homomorphic image h[C].Being T zS -closed, the semigroup h[C] is zero-closed, see Lemma 2.1.By Theorem 2.3(1), the zero-closed countable semigroup h[C] is polybounded and by Lemma 2.2, h[C] is polyfinite.By the preceding statement, the semigroup X is A-centrobounded.4. Assume that the semigroup X is absolutely T zS -closed, |h[C]| ≤ cov(M), and for every e ∈ A the subsemigroup Ce ⊆ H e is countable.Since h[C] ⊆ e∈A Ce ⊆ H , the subsemigroup h[C] of the metric group (H , ρ) is separable.By Lemma 2.4, the prime coideal C is absolutely T zS -closed and so is its homomorphic image h[C].By the absolute T zS -closedness of h[C], the semigroup h[C] is zero-closed and also h[C] is closed in the zero-dimensional topological group H . Then the metric ρ h[C]×h[C] is complete and hence h[C] is a Polish space.We claim that h[C] is polybounded.If |h[C]| < c, then the Polish space h[C] is countable, see [46, 6.5].By Theorem 2.3(1), the countable zero-closed semigroup h[C] is polybounded.If |h[C]| = c, then the inequality |h[C]| ≤ cov(M) implies cov(M) = c and then h[C] is polybounded by Theorem 2.3(2).So, in both cases, the semigroup h[C] is polybounded.By Lemma 2.2, h[C] is polyfinite.By the second statement of this theorem, the semigroup X is A-centrobounded.
5. Assume that |h[C]| ≤ c and for every e ∈ A the semigroup Ce ⊆ H e of X is bounded.For a bounded subset B ⊆ X , let exp(B) def e n = Z (X ) ∩ A, and z n = (x + n e n )(x − n e n )−1 ∈ Z (X ) ∩ H e n ; (ii) z i n = e n for every 1 ≤ i ≤ nμ n where μ n Consider the group H = n∈ω H e n , and the homomorphism h : C → H , h : x → (xe n ) n∈ω .Observe that h = pr • h, where pr : H → H , pr : (x a ) a∈A → (x e n ) n∈ω , is the projection.For every n ∈ ω, let pr n : H → H e n , pr n : (x k ) k∈ω → x n , be the n-th coordinate projection.Endow the group H with the complete invariant metricρ : H × H → R, ρ ((x n ) n∈ω , (y n ) n∈ω ) → {0} ∪ 1 2 n : n ∈ ω, x n = y n .By Lemma 2.4, the prime coideal C of the absolutely T zS -closed semigroup X is absolutely T zS -closed and so is its homomorphic image h [C] ⊆ H . Since the Tychonoff product topology on the group H = n∈ω H e n is zero-dimensional, the absolutely T zS -closed subsemigroup h [C] of H is closed in H . Being absolutely T zS -closed, the semigroup h[C] def = lcm{exp(Ce k ) : k < n}.