On subgroups of saturated or totally bounded paratopological groups

A paratopological group G is saturated if the inverse U ⁻¹ of each non-empty set U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous b...

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Published in:	Algebra and Discrete Mathematics
Date:	2003
Main Authors:	Banakh, T., Ravsky, S.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2003
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Cite this:	On subgroups of saturated or totally bounded paratopological groups / T. Banakh, S. Ravsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 1–20. — Бібліогр.: 25 назв. — англ.

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spelling	Banakh, T. Ravsky, S. 2019-06-17T11:14:56Z 2019-06-17T11:14:56Z 2003 On subgroups of saturated or totally bounded paratopological groups / T. Banakh, S. Ravsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 1–20. — Бібліогр.: 25 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 22A15, 54H10, 54H11. https://nasplib.isofts.kiev.ua/handle/123456789/155719 A paratopological group G is saturated if the inverse U ⁻¹ of each non-empty set U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group G [such that for each neighborhood U ⊂ H of the unit e there is a closed subset F ⊂ G with e ∈ h ⁻¹ (F) ⊂ U]. As an application we construct a paratopological group whose character exceeds its π-weight as well as the character of its group reflexion. Also we present several examples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups. The first author was supported in part by the Slovenian-Ukrainian research grant SLO-UKR 02-03/04. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On subgroups of saturated or totally bounded paratopological groups Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	On subgroups of saturated or totally bounded paratopological groups
spellingShingle	On subgroups of saturated or totally bounded paratopological groups Banakh, T. Ravsky, S.
title_short	On subgroups of saturated or totally bounded paratopological groups
title_full	On subgroups of saturated or totally bounded paratopological groups
title_fullStr	On subgroups of saturated or totally bounded paratopological groups
title_full_unstemmed	On subgroups of saturated or totally bounded paratopological groups
title_sort	on subgroups of saturated or totally bounded paratopological groups
author	Banakh, T. Ravsky, S.
author_facet	Banakh, T. Ravsky, S.
publishDate	2003
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	A paratopological group G is saturated if the inverse U ⁻¹ of each non-empty set U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group G [such that for each neighborhood U ⊂ H of the unit e there is a closed subset F ⊂ G with e ∈ h ⁻¹ (F) ⊂ U]. As an application we construct a paratopological group whose character exceeds its π-weight as well as the character of its group reflexion. Also we present several examples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/155719
citation_txt	On subgroups of saturated or totally bounded paratopological groups / T. Banakh, S. Ravsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 1–20. — Бібліогр.: 25 назв. — англ.
work_keys_str_mv	AT banakht onsubgroupsofsaturatedortotallyboundedparatopologicalgroups AT ravskys onsubgroupsofsaturatedortotallyboundedparatopologicalgroups
first_indexed	2025-11-25T21:28:45Z
last_indexed	2025-11-25T21:28:45Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2003). pp. 1 – 20 c© Journal “Algebra and Discrete Mathematics” On subgroups of saturated or totally bounded paratopological groups Taras Banakh and Sasha Ravsky Communicated by M. Ya. Komarnytskyj Dedicated to R. I. Grigorchuk on the occasion of his 50th birthday Abstract. A paratopological group G is saturated if the in- verse U−1 of each non-empty set U ⊂ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group G [such that for each neighborhood U ⊂ H of the unit e there is a closed subset F ⊂ G with e ∈ h−1(F ) ⊂ U ]. As an application we construct a paratopological group whose character exceeds its π-weight as well as the character of its group reflexion. Also we present several ex- amples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups. In this paper we continue investigations of paratopological groups, started by the authors in [Ra1], [Ra2], [BR1]. By a paratopological group we understand a pair (G, τ) consisting of a group G and a topology τ on G making the group operation · : G × G → G of G continuous (such a topology τ will be called a semigroup topology on G). If, in addition, the operation (·)−1 : G → G of taking the inverse is continuous with respect to the topology τ , then (G, τ) is a topological group. All topological spaces considered in this paper are supposed to be Hausdorff if the opposite is not stated. The first author was supported in part by the Slovenian-Ukrainian research grant SLO-UKR 02-03/04. 2000 Mathematics Subject Classification: 22A15, 54H10, 54H11. Key words and phrases: saturated paratopological group, group reflexion. Jo u rn al A lg eb ra D is cr et e M at h .2 On subgroups of saturated paratopological groups The absence of the continuity of the inverse in paratopological groups results in appearing various pathologies impossible in the category of topological groups, which makes the theory of paratopological groups quite interesting and unpredictable. In [Gu] I. Guran has introduced a relatively narrow class of so-called saturated paratopological groups which behave much like usual topological groups. Following I.Guran we define a paratopological group G to be saturated if the inverse U−1 of any nonempty open subset U of G has non-empty interior in G. A stan- dard example of a saturated paratopological group with discontinuous inverse is the Sorgenfrey line, that is the real line endowed with the Sor- genfrey topology generated by the base consisting of half-intervals [a, b), a < b. Important examples of saturated paratopological groups are to- tally bounded groups, that is paratopological groups G such that for any neighborhood U ⊂ G of the origin in G there is a finite subset F ⊂ G with G = FU = UF , see [Ra3, Proposition 3.1]. Observing that each subgroup of the Sorgenfrey line is saturated, I.Guran asked if the same is true for any saturated paratopological group. This question can be also posed in another way: which paratopological groups embed into saturated ones? A similar question concerning em- bedding into totally bounded semi- or paratopological groups appeared in [AH]. In this paper we shall show that the necessary and sufficient condition for a paratopological group G to embed into a saturated (totally bounded) paratopological group is the existence of a bijective continuous group homomorphism h : G → H onto a topological (totally bounded) group H. The latter property of G will be referred to as the [-separateness (resp. Bohr separateness). [-Separated paratopological groups can be equivalently defined with help of the group reflexion of a paratopological group. Given a paratopo- logical group G let τ [ be the strongest group topology on G, weaker than the topology of G. The topological group G[ = (G, τ [), called the group reflexion of G, has the following characteristic property: the identity map i : G → G[ is continuous and for every continuous group homomor- phism h : G → H from G into a topological group H the homomorphism h ◦ i−1 : G[ → H is continuous. According to [BR1], a neighborhood base of the unit of the group reflexion G[ of a saturated (more generally, 2-oscillating) paratopological group G consists of the sets UU−1 where U runs over neighborhoods of the unit in G. For instance, the group reflexion of the Sorgenfrey line is the usual real line. There is also a dual notion of a group co-reflexion. Given a paratopo- logical group G let τ] be the weakest group topology on G, stronger than the topology of G. The topological group G] = (G, τ]) is called the group Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 3 co-reflexion of G. According to [Ra1], a neighborhood base of the unit of the group co-reflexion G] of a paratopological group G consists of the sets U ∩ U−1 where U runs over neighborhoods of the unit in G. A paratopological group is called ]-discrete provided its group co-reflection is discrete. For instance, the Sorgenfrey line is ]-discrete. A subset A of a paratopological group G will be called [-closed if A is closed in the topology τ [. A paratopological group G is called [-separated provided its group reflexion G[ is Hausdorff; G is called [-regular if each neighborhood U of the unit e of G contains a [-closed neighborhood of e. It is easy to see that each [-regular paratopological group is regular and [-separated, see [BR1]. For saturated groups the converse is also true: each regular saturated paratopological group is [-regular, see [BR1, Theorem 3]. The notions of a [-separated (resp. [-regular) paratopological group is a partial case of a more general notion of a G-separated (resp. G-regular) paratopological group, where G is a class of topological groups. Namely, we call a paratopological group G to be • G-separated if G admits a continuous bijective homomorphism h : G → H onto a topological group H ∈ G; • G-regular if G admits a regular continuous homomorphism h : G → H onto a topological group H ∈ G. A continuous map h : X → Y between topological spaces is defined to be regular if for each point x ∈ X and a neighborhood U of x in X there is a closed subset F ⊂ Y such that h−1(F ) is a closed neighborhood of x with h−1(F ) ⊂ U . As we shall see later, any injective continuous map from a kω-space is regular. We remind that a topological space X is defined to be a kω-space if there is a countable cover K of X by compact subsets of X, determining the topology of X in the sense that a subset U of X is open in X if and only if the intersection U∩K is open in K for any compact set K ∈ K. According to [FT], each Hausdorff kω-space is normal. Under a (para)topological kω-group we shall understand a (para)topological group whose underlying topological space is kω. Many examples of kω-spaces appear in topological algebra as free objects in various categories, see [Cho]. In particular, the free (abelian) topological group over a compact Hausdorff space is a topological kω-group, see [Gra]. Proposition 1. Any injective continuous map f : X → Y from a Haus- dorff kω-space X into a Hausdorff space Y is regular. Jo u rn al A lg eb ra D is cr et e M at h .4 On subgroups of saturated paratopological groups Proof. Fix any point x ∈ X and an open neighborhood U ⊂ X of x. Let {Kn} be an increasing sequence of compacta determining the topology of the space X. Without loss of generality, we may assume that K0 = {x}. Let V0 = K0 and W0 = Y . By induction, for every n ≥ 1 we shall find an open neighborhood Vn of x in Kn and an open neighborhood Wn of f(V n) in Y such that 1) f−1(Wn) ∩ Kn ⊂ U ; 2) V n ⊂ U ∩ ⋂ i≤n f−1(Wi); 3) V n ⊂ Vn+1. Assume that for some n the sets Vi, Wi, i < n, have been constructed. It follows from (2) that f(V n−1) and f(Kn \U) are disjoint compact sets in the Hausdorff space Y . Consequently, the compact set f(V n−1) has an open neighborhood Wn ⊂ Y whose closure Wn misses the compact set f(Kn \U). Such a choice of Wn and the condition (2) imply that V n−1 ⊂ U ∩ ⋂ i≤n f−1(Wi). Using the normality of the compact space Kn find an open neighborhood Vn of V n−1 in Kn such that V n ⊂ U ∩ ⋂ i≤n f−1(Wi), which finishes the inductive step. It is easy to see that V = ⋃ n∈ω Vn is an open neighborhood of x such that f−1(f(V )) ⊂ f−1( ⋂ i∈ω W i) ⊂ U , which just yields the regularity of the map f . For paratopological kω-groups this proposition yields the equivalence between the G-regularity and G-separateness. Corollary 1. Let G be a class of topological groups. A paratopological kω-group is G-separated if and only if it is G-regular. A class G of topological groups will be called • closed-hereditary if each closed subgroup of a group G ∈ G belongs to the class G; • H-stable, where H is a topological group, if G × H ∈ G for any topological group G ∈ G. For a topological space X by χ(X) we denote its character, equal to the smallest cardinal κ such that each point x ∈ X has a neighborhood base of size ≤ κ. Now we are able to give a characterization of subgroups of saturated paratopological groups (possessing certain additional properties). Theorem 1. Suppose T is a saturated ]-discrete nondiscrete paratopo- logical group and G is a closed-hereditary T [-stable class of topological groups. A paratopological group H is G-separated if and only if H is a Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 5 [-closed subgroup of a saturated paratopological group G with G[ ∈ G, χ(G) = max{χ(H), χ(T )}, and \|G/H\| = \|T \|. A similar characterization holds for G-regular paratopological groups. We remind that a paratopological group G is called a paratopological SIN- group if any neighborhood U of the unit e in G contains a neighborhood W ⊂ G of e such that gWg−1 ⊂ U for all g ∈ G. It is easy to check that every paratopological SIN-group has a base at the unit consisting of invariant open neighborhoods, see [Ra3, Ch.4] (as expected, a subset A of a group G is called invariant if xAx−1 = A for all x ∈ G). Finally we define a notion of a Sorgenfrey paratopological group which crystallizes some important properties of the Sorgenfrey topology on the real line. A paratopological group G is defined to be Sorgenfrey if G is non-discrete, saturated and contains a neighborhood U of the unit e such that for any neighborhood V ⊂ G of e there is a neighborhood W ⊂ G of e such that x, y ∈ V for any elements x, y ∈ U with xy ∈ W . Observe that each Sorgenfrey paratopological group is ]-discrete. Theorem 2. Suppose T is a first countable saturated regular Sorgenfrey paratopological SIN-group and G is a closed-hereditary T [-stable class of first countable topological SIN-groups. A paratopological group H is G- regular if and only if H is a [-closed subgroup of a saturated [-regular paratopological group G with G[ ∈ G, χ(G) = χ(H), and \|G/H\| = \|T \|. Theorem 2 in combination with Corollary 1 yield Corollary 2. Suppose T is a first countable saturated regular Sorgenfrey paratopological SIN-group and G is a closed-hereditary T [-stable class of first countable topological SIN-groups. A paratopological kω-group H is G- separated if and only if H is a [-closed subgroup of a saturated [-regular paratopological group G with G[ ∈ G, χ(G) = χ(H), and \|G/H\| = \|T \|. As we said, any (regular) saturated paratopological group is [-sepa- rated (and [-regular), see [BR1]. Observe that a paratopological group G is [-separated if and only if G is TopGr-separated where TopGr stands for the class of all Hausdorff topological groups. These observations and Theorem 1 imply Corollary 3. A paratopological group H is [-separated if and only if H is a ([-closed) subgroup of a saturated paratopological group. Unfortunately we do not know the answer to the obvious [-regular version of the above corollary. Jo u rn al A lg eb ra D is cr et e M at h .6 On subgroups of saturated paratopological groups Problem 1. Is every [-regular paratopological group a subgroup of a reg- ular saturated paratopological group? For first-countable paratopological SIN-groups the answer to this pro- blem is affirmative. Corollary 4. A first-countable paratopological SIN-group H is [-regular if and only if H is a [-closed subgroup of a regular first-countable saturated paratopological SIN-group G with \|G/H\| = ℵ0. Proof. Taking into account that H is a first-countable paratopological SIN-group and applying [BR1, Proposition 3], we conclude that H[ is a first-countable topological SIN-group. Let T be the quotient group Q/Z of the group of rational numbers, endowed with the Sorgenfrey topol- ogy generated by the base consisting of half-intervals. Observe that T is a [-regular Sorgenfrey abelian paratopological group and the class of first countable topological SIN-groups is T [ stable, closed-hereditary and contains H[. Now to finish the proof apply Theorem 2. Next, we apply Theorems 1 and 2 to the class TBG (resp. FCTBG) of (first countable) totally bounded topological groups. We remind that a paratopological group G is totally bounded if for any neighborhood U of the unit in G there is a finite subset F ⊂ G with UF = FU = G. It is known that each totally bounded paratopological group is saturated and a saturated paratopological group G is totally bounded if and only if its group reflexion G[ is totally bounded, see [Ra2], [BR1]. An example of a [-regular totally bounded Sorgenfrey paratopological group is the circle T = {z ∈ C : \|z\| = 1} endowed with the Sorgenfrey topology generated by the base consisting of half-intervals {z ∈ T : a ≤ Arg(z) < b} where 0 ≤ a < b ≤ 2π. We define a paratopological group G to be Bohr separated (resp. Bohr regular, fcBohr regular) if it is TBG-separated (resp. TBG-regular, FCTBG-regular). In this terminology Theorem 1 implies Corollary 5. A paratopological group H is Bohr separated if and only if H is a ([-closed) subgroup of a totally bounded paratopological group. It is interesting to compare Corollary 5 with another characterizing theorem supplying us with many pathological examples of pseudocom- pact paratopological groups. We remind that a topological space X is pseudocompact if each locally finite family of non-empty open subsets of X is finite. It should be mentioned that a Tychonoff space X is pseu- docompact if and only if each continuous real-valued function on X is bounded. Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 7 Theorem 3. An abelian paratopological group H is Bohr separated if and only if H is a subgroup of a pseudocompact abelian paratopological group G with χ(G) = χ(H). The following characterization of fcBohr regular paratopological gro- ups follows from Theorem 2 applied to the class G = FCTBG and the quotient group T = Q/Z endowed with the standard Sorgenfrey topology. Corollary 6. A paratopological group H is fcBohr regular if and only if H is a [-closed subgroup of a regular totally bounded paratopological group G with ℵ0 = χ(G[) ≤ χ(G) = χ(H) and \|G/H\| ≤ ℵ0. In some cases the fcBohr (= FCTBG) regularity is equivalent to the Bohr (= TBG) regularity. We recall that a topological space X has countable pseudocharacter if each one point subset of X is a Gδ-subset. Proposition 2. A Bohr regular paratopological group H is fcBohr regular provided one of the following conditions is satisfied: 1. H is a kω-space with countable pseudocharacter; 2. H is first countable and Lindelöf. Proof. Using the Bohr regularity of H, find a regular bijective continuous homomorphism h : H → K onto a totally bounded topological group K. Denote by eH and eK the units of the groups H, K, respectively. 1. First assume that H is a kω-space with countable pseudo-character. In this case the set H \ {eH} is σ-compact as well as its image h(H \ {eH}) = K\{eK}. It follows that the totally bounded group K has count- able pseudo-character. Now it is standard to find a bijective continuous homomorphism i : K → G of K onto a first countable totally bounded topological group G, see [Tk, 4.5]. Since the composition f ◦ h : H → G is bijective, the group H is FCTBG-separated and by Proposition 1 is fcBohr regular. 2. Next assume that H is first-countable and Lindelöf. Fix a sequence (Un)n∈ω of open subsets of H forming a neighborhood base at eH . For every n ∈ ω fix a closed neighborhood Wn ⊂ Un whose image h(Wn) is closed in K. Let us call an open subset U ⊂ K cylindrical if U = f−1(V ) for some continuous homomorphism f : K → G into a first countable compact topological group G and some open set V ⊂ G. It follows from the total boundedness of K that open cylindrical subsets form a base of the topology of the group K, see [Tk, 3.4]. Using the Lindelöf property of the set f(H \ Un), for every n ∈ ω find a countable cover Un of f(H \ Un) by open cylindrical subsets such Jo u rn al A lg eb ra D is cr et e M at h .8 On subgroups of saturated paratopological groups that ∪Un ∩ h(Wn) = ∅. Then U = ⋃ n∈ω Un is a countable collection of open cylindrical subsets and we can produce a continuous homomorphism f : K → G onto a first countable totally bounded topological group such that each set U ∈ U is the preimage U = f−1(V ) of some open set V ⊂ K. To finish the proof it rests to observe that the composition f ◦ h : H → G is a regular bijection of H onto a first countable totally bounded topological group. It can be shown that the character of any non-locally compact parato- pological kω-group with countable pseudo-character is equal to the small cardinal d, well-known in the Modern Set Theory, see [JW], [Va]. By definition, d is equal to the cofinality of Nω endowed with the natural partial order: (xi)i∈ω ≤ (yi)i∈ω iff xi ≤ yi for all i. More precisely, d is equal to the smallest size of a subset C ⊂ Nω cofinal in the sense that for any x ∈ Nω there is y ∈ C with y ≥ x. It is easy to see that ℵ1 ≤ d ≤ c. The Martin Axiom implies d = c. On the other hand, there are models of ZFC with d < c, see [Va]. We shall apply Corollary 6 to construct examples of paratopological groups whose character exceed their π-weight as well as the character of their group reflexions. We recall that the π-weight πw(X) of a topolog- ical space X is the smallest size of a π-base, i.e., a collection B of open nonempty subsets of X such that each nonempty open subset U of X contains an element of the family B. According to [Tk, 4.3] the π-weight of each topological group coincides with its weight. Corollary 7. For any uncountable cardinal κ ≤ d there is a [-regular to- tally bounded countable abelian paratopological group G with ℵ0 = χ(G[) = πw(G) < χ(G) = κ. Proof. Take any non-metrizable countable abelian FCTBG-separated to- pological kω-group (H, τ) (for example, let H be a free abelian group over a convergent sequence, see [FT]). The group H, being FCTBG- separated, admits a bijective continuous homomorphism h : H → K onto a first countable totally bounded abelian topological group K. By Proposition 1 this homomorphism is regular. According to [FT, 22)] or [Ban], the space H contains a copy of the Fréchet-Urysohn fan Sω and thus has character χ(H) ≥ χ(Sω) ≥ d. Using this fact and the countability of H, by transfinite induction (over ordinals < κ) construct a group topology σ ⊂ τ on H such that χ(H, σ) = κ and the homomorphism h : (H, σ) → K is regular. This means that the group (H, σ) is fcBohr regular. Now we can apply Corollary 6 to embed the group (H, σ) into a totally bounded countable abelian paratopological group G such that ℵ0 = χ(G[) < χ(G) = χ(H, σ) = κ. Since G is Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 9 saturated and abelian, [-open subsets of G form a π-base which implies πw(G) = ω(G[) = ℵ0. Remark 1. It is interesting to compare Corollary 7 with a result of [BRZ] asserting that there exists a [-regular countable paratopological group G with ℵ0 = χ(G) < χ(G[) = d. Such a paratopological group G cannot be saturated since χ(G[) ≤ χ(G) for any saturated (more generally, any 2-oscillating) paratopological group G, see [BR1]. We finish our discussion with presenting examples of regular (para)to- pological groups which embed into totally bounded paratopological gro- ups but fail to embed into regular totally bounded paratopological groups. For that it suffices to find a Bohr separated group which is not Bohr regular. Let us remark that each locally convex linear topological space (or more generally each locally quasi-convex abelian topological group, see [A] or [Ba]) is Bohr regular. On the other hand, there exist (non-locally convex) linear metric spaces which fail to be Bohr separated or Bohr regular. The simplest example can be constructed as follows. Con- sider the linear space C[0, 1] of continuous real-valued functions on the closed interval [0, 1] and endow it with the invariant metrics d1/2(f, g) = ∫ 1 0 √ \|f(t) − g(t)\|dt, p(f, g) = ∑ n∈ω min{2−n, \|f(tn)−g(tn)\|} and ρ(f, g) = d1/2(f, g) + p(f, g) where {tn : n ∈ ω} is an enumeration of ra- tional numbers of [0, 1]. It is well-known that the linear metric space (C[0, 1], d1/2) admits no non-zero linear continuous functional and fails to be Bohr separated. The linear metric space (C[0, 1], ρ) is even more interesting. We remind that an abelian group G is divisible (resp. torsion- free) if for any a ∈ G and natural n the equation xn = a has a solution (resp. has at most one solution) x ∈ G. Proposition 3. The linear metric space L = (C[0, 1], ρ) is Bohr sep- arated but fails to be Bohr regular. Moreover, L is a [-closed subgroup of a totally bounded abelian torsion-free divisible group, but fails to be a subgroup of a regular totally bounded paratopological group. Proof. The Bohr separatedness of L follows from the continuity of the maps χn : L → R, χn : f 7→ f(tn), for n ∈ ω. Let us show that the group L fails to be Bohr regular. For this we first prove that each linear continuous functional ψ : (L, ρ) → R is continuous with respect to the “product” metric p. Consider the open convex subset C = ψ−1(−1, 1) of L. By the continuity of ψ, there are n ≥ 1 and ε > 0 such that x ∈ C for any x ∈ L with d1/2(x, 0) < ε and \|x(ti)\| < ε for all i ≤ n. Let L0 = {x ∈ L : x(ti) = 0 for Jo u rn al A lg eb ra D is cr et e M at h .10 On subgroups of saturated paratopological groups all i ≤ n} and observe that the convex set C∩L0 contains the open ε-ball with respect to the restriction of the metric d1/2 on L0. Now the standard argument (see [Ed, 4.16.3]) yields C ∩ L0 = L0 and L0 ⊂ ⋂ k≥1 1 kC = ψ−1(0). Hence the functional ψ factors through the quotient space L/L0 and is continuous with respect to the metric p (this follows from the continuity of the quotient homomorphism L → L/L0 with respect to p). If χ : L → T is any character on L (that is a continuous group homo- morphism into the circle T = R/Z), then it is easy to find a continuous linear functional ψ : L → R such that χ = π ◦ ψ, where π : R → T is the quotient homomorphism. As we have already shown, the functional ψ is continuous with respect to the metric p and so is the character χ. Finally, we are able to prove that the group L fails to be Bohr reg- ular. Assuming the converse we would find a continuous regular homo- morphism h : L → H onto a totally bounded abelian topological group H. The group H, being abelian and totally bounded, is a subgroup of the product Tκ for some cardinal κ, see [Mo]. Then the above discus- sion yields that h is continuous with respect to the metric p. In this situation the regularity of h implies the regularity of the identity map (L, ρ) → (L, p). But this map certainly is not regular: for any 2−n-ball B = {x ∈ L : ρ(x, 0) < 2−n} its closure in the metric p contains the linear subspace {x ∈ L : x(ti) = 0 for all i ≤ n} and thus lies in no ball. Therefore the group L is Bohr separated but not Bohr regular. Let G be the class of all totally bounded abelian divisible torsion- free topological groups. The group L, being Bohr separated, abelian, divisible, and torsion-free, is G-separated. Pick any irrational number α ∈ T = R/Z and consider the subgroup T = {qα : q ∈ Q} of the circle T endowed with the Sorgenfrey topology. It is clear that T is a totally bounded ]-discrete paratopological group with T [ ∈ G. By Theorem 1, L is a [-closed subgroup of a saturated paratopological groups G with G[ ∈ G which implies that G is totally bounded abelian, divisible and torsion-free. On the other hand, L admits no embedding into a regular totally bounded paratopological group G. Indeed, assuming that L ⊂ G is such an embedding, apply Theorem 3 of [BR1] to conclude that the identity homomorphism id : G → G[ is regular and so is its restriction id\|L, which would imply the Bohr regularity of L. There is also an alternative method of constructing Bohr separated but not Bohr regular paratopological groups, based on the concept of a Lawson paratopological group. Following [BR1] we define a paratopo- logical group G to be Lawson if it has a neighborhood base at the unit consisting of subsemigroups of G. According to [BR1] there is a regular Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 11 Lawson paratopological group failing to be [-separated. On the other hand, there are Lawson paratopological groups which are [-regular and Bohr separated but are not topological groups, see Example 2 [BR1] or Example 1 below. We shall show that a [-regular paratopological group G is a topological group provided its group reflexion G[ is topologically periodic. We remind that a paratopological group G is topologically pe- riodic if for each x ∈ G and a neighborhood U ⊂ G of the unit there is a number n ≥ 1 such that xn ∈ U , see [BG]. It is easy to show that each totally bounded topological group is topologically periodic. For paratopological groups it is not true: according to Theorem 2 there is a [-regular totally bounded paratopological group G which contains the discrete group Z of integers and thus cannot be topologically periodic. The class of topologically periodic topological groups will be denoted by TPTG. Proposition 4. Each TPTG-regular Lawson paratopological group is a topological group. Proof. Let (G, τ) be a Lawson paratopological group and σ ⊂ τ be a topology turning G into a topologically periodic topological group such that (G, τ) has a base B at the unit consisting of subsemigroups, closed in the topology σ. We are going to show that an arbitrary element U ∈ B is in fact a subgroup of G. For this purpose suppose that there exists an element x ∈ U−1\U . Then x−1 ∈ U and xm ∈ U for all m < 0 because U is a subsemigroup of G. Since the set U is closed in the topology σ, there exists a neighborhood V ∈ σ of unit such that xV ∩ U = ∅. By the topological periodicity of (G, σ), there exists a number n < −1 with xn ⊂ V . Then xn+1 ∩ U = ∅ which is a contradiction. Since each totally bounded topological group is topologically periodic this Proposition implies Corollary 8. Each Bohr regular Lawson paratopological group is a topo- logical group. On the other hand, abelian Lawson paratopological groups are Bohr separated. Proposition 5. Each abelian Lawson paratopological group is Bohr sep- arated. Proof. Let G be such the group. Then G[ has a neighborhood base B at the unit, consisting of subgroups. For every group H ∈ B the group G/H, being abelian and discrete, is Bohr separated [Mo]. Since the family {G → G/H : H ∈ B} of quotient maps separates the points of the group G, the group G is Bohr separated too. Jo u rn al A lg eb ra D is cr et e M at h .12 On subgroups of saturated paratopological groups Corollary 8 and Proposition 5 allow us to construct simple examples of Bohr separated Lawson paratopological groups which are not Bohr regular. Example 1. There is a countable [-regular saturated Lawson paratopo- logical abelian group H which is Bohr separated but not Bohr regular. The group H has the following properties: 1. H is a [-closed subgroup of a countable first-countable abelian to- tally bounded paratopological group; 2. H is a [-closed subgroup of a first-countable abelian pseudocompact paratopological group; 3. H fails to be a subgroup of a regular totally bounded (or pseudo- compact) paratopological group. Proof. Consider the direct sum Zω 0 = {(xi)i∈ω ∈ Zω : xi = 0 for all but finitely many indices i} of countably many copies of the group Z of integers. Endow the group Zω 0 with a shift invariant topology τ whose neighborhood base at the origin consists of the sets Un = {0}∪ ⋃ m≥n Wm where Wm = {(xi)i∈ω ∈ Zω 0 : xi = 0 for all i < m and xm > 0} for m ≥ 0. It is easy to see that H = (Zω 0 , τ) is a [-regular countable first-countable saturated Lawson paratopological group which is not a topological group. By Proposition 5 and Corollary 8 the group H is Bohr separated but not Bohr regular. By Theorem 1, H is a [-closed subgroup of a first-countable totally bounded countable paratopological group and by Theorem 3, H is a [- closed subgroup of a first-countable abelian pseudocompact paratopolog- ical group. Assuming that H is a subgroup of a regular totally bounded or pseu- docompact paratopological group G and applying Theorem 3 of [BR1] and [RR] we would get that both G and H are Bohr regular which is impossible. In the proofs of our principal results we shall often exploit the follow- ing characterization of semigroup topologies on groups from [Ra1, 1.1]. Lemma 1. A family B of subsets containing a unit e of a group G is a neighborhood base at e of some semigroup topology τ on G if and only if B satisfies the following four Pontryagin conditions: 1. (∀U, V ∈ B)(∃W ∈ B) : W ⊂ U ∩ V ; 2. (∀U ∈ B)(∃V ∈ B) : V 2 ⊂ U ; 3. (∀U ∈ B)(∀x ∈ U)(∃V ∈ B) : xV ⊂ U ; Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 13 4. (∀U ∈ B)(∀x ∈ G)(∃V ∈ B) : x−1V x ⊂ U . The topology τ is Hausdorff if and only if 5. ⋂ {UU−1 : U ∈ B} = {e}. 1. Proof of Theorem 1 The necessity is evident. We shall prove the sufficiency. Let (H, τ) be a G- separated paratopological group, where G is T [-stable class of topological groups. Since the group H is G-separated, there exists a group topology σ on the group H such that (H, σ) ∈ G. We shall define the topology on the product G = H × T as follows. Let Bτ , Bσ and BT be open bases at the unit of the groups (H, τ), (H, σ) and T respectively. For arbitrary neighborhoods Uτ ∈ Bτ , Uσ ∈ Bσ and UT ∈ BT with Uτ ⊂ Uσ put [Uτ , Uσ, UT ] = Uτ ×{eT }∪Uσ×(UT \{eT }), where eH and eT are the units of the groups H and T respectively. The family of all such [Uτ , Uσ, UT ] will be denoted by B. Now we verify the Pontryagin conditions for the family B. The Condition 1 is trivial. To check Condition 2 consider an arbitrary set [Uτ , Uσ, UT ] ∈ B. There exist neighborhoods Vτ ∈ Bτ , Vσ ∈ Bσ such that V 2 τ ⊂ Uτ , V 2 σ ⊂ Uσ and Vτ ⊂ Vσ. Since the group T# is discrete then there is a neighborhood VT ⊂ BT such that (VT \{eT }) 2 ⊂ UT \{eT }. Then [Vτ , Vσ, VT ]2 ⊂ [Uτ , Uσ, UT ]. To verify Condition 3 consider an arbitrary point x ∈ [Uτ , Uσ, UT ] ∈ B. If x = (xH , eT ), where xH ∈ Uτ then there exist neighborhoods Vτ ∈ Bτ , Vσ ∈ Bσ such that Vτ ⊂ Vσ, xHVτ ⊂ Uτ and xHVσ ⊂ Uσ. Then x[Vτ , Vσ, UT ] ⊂ [Uτ , Uσ, UT ]. If x = (xH , xT ), where xH ∈ Uσ and xT ∈ UT \{eT } then there exist neighborhoods Vτ ∈ Bτ , Vσ ∈ Bσ and VT ∈ BT such that Vτ ⊂ Vσ, xHVσ ⊂ Uσ and xT VT ⊂ UT \{eT }. Then x[Vτ , Vσ, VT ] ⊂ [Uτ , Uσ, UT ]. Condition 4. Let x = (xH , xT ) ⊂ H × T be an arbitrary point. Then there are neighborhoods Vτ ∈ Bτ , Vσ ∈ Bσ and VT ∈ BT such that Vτ ⊂ Vσ, x−1 H VτxH ⊂ Uτ , x−1 H VσxH ⊂ Uσ and x−1 T VT xT ⊂ UT . Then x−1[Vτ , Vσ, VT ]x ⊂ [Uτ , Uσ, UT ]. Hence the family B is a base of a semigroup topology on the group G. Denote this semigroup topology by ρ. The inclusion ⋂ {[Uτ , Uσ, UT ] · [Uτ , Uσ, UT ]−1 : Uτ ∈ Bτ , Uσ ∈ Bσ, UT ∈ BT } ⊂ {UσU−1 σ × UT U−1 T : Uσ ∈ Bσ, UT ∈ BT } = {(eH , eT )} implies that the topology ρ is Hausdorff. Since the groups T and (H, σ) are saturated and the group T is nondis- crete, the group (G, ρ) is saturated too. According to [BR1, Proposition 3] the base at the unit of the topology ρ[ consists of the sets UU−1, where U ∈ B. Thus the topology ρ[ coincides with the product topology of the Jo u rn al A lg eb ra D is cr et e M at h .14 On subgroups of saturated paratopological groups groups (H, σ) × T [ and hence (G, ρ[) ∈ G and H is a [-closed subgroup of the group G. 2. Proof of Theorem 2 The “if” part of Theorem 2 is trivial. To prove the “only if” part, suppose that T and (H, τ) are paratopological groups with the units eT and eH , satisfying the hypothesis of Theorem 2. Using the Sorgenfrey property of the group T , choose an open in- variant neighborhood U0 of the unit eT such that for any neighborhood U ⊂ T of eT there is a neighborhood U ′ ⊂ T of eT such that x, y ∈ U for any elements x, y ∈ U0 with xy ∈ U ′. By induction we can build a sequence {Un : n ∈ ω} of invariant open neighborhoods of eT satisfying the following conditions: (1) {Un : n ∈ ω} is a neighborhood base at the unit eT of the group T ; (2) U2 n+1 ⊂ Un for every n ∈ ω; (3) for every n ∈ ω and any points x, y ∈ U0 the inclusion xy ∈ Un+1 implies x, y ∈ Un; (4) Un [ $ Un−1 for every n ∈ ω, where Un [ denotes the closure of the set Un in the topology of T [. Remark that the condition (3) yields (5) (U0\Un)U0 ∩Un+1 = ∅ and hence U0\Un ∩Un+1U −1 0 = ∅ for all n. Since the group T is saturated, we can apply Proposition 3 of [BR1] to conclude that the set Un+2U −1 0 is a neighborhood of the unit in T [. Then the set Un+2Un+2U −1 0 ⊂ Un+1U −1 0 is a neighborhood of Un+2 in T [. This observation together with (5) yields (6) U0\Un [ ∩ Un+2 = ∅ for all n. It follows from our assumptions on (H, τ) that there exists a group topology σ ⊂ τ on H such that the group (H, σ) belongs to the class G and (H, τ) has a neighborhood base Bτ at the unit eH consisting of sets, closed in the topology σ. By induction we can build a base {Vn : n ∈ ω} of open symmetric invariant neighborhoods of eH in the topology σ such that V 2 n+1 ⊂ Vn for every n ∈ ω. Consider the product H × T and identify H with the subgroup H × {eT } of H × T . It rests to define a topology on H × T . At first we shall introduce an auxiliary sequence {Wn} of “neighborhoods” of (eH , eT ) satisfying the Pontryagin Conditions 1,2, and 4. For every n ∈ ω let Wn = {(eH , eT )} ∪ ⋃ i>2n Vni × (Ui−1 \ Ui) (*) and observe that Wn+1 ⊂ Wn for all n. Let us verify the Pontryagin Conditions 1,2,4 for the sequence (Wn). Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 15 To verify Conditions 1 and 2 it suffices to show that W 2 n ⊂ Wn−1 for all n ≥ 1. Fix any elements (x, t), (x′, t′) ∈ Wn. We have to show that (xx′, tt′) ∈ Wn−1. Without loss of generality, we can assume that t, t′ 6= eT . In this case we may find numbers i, i′ > 2n with (x, t) ∈ Vni × (Ui−1\Ui) and (x′, t′) ∈ Vni′ × (Ui′−1\Ui′). For j = min{i, i′} the Conditions (2), (5) imply (xx′, tt′) ∈ Vnj−1 × (Uj−2\Uj+1) ⊂ j+1 ⋃ k=j−1 V(n−1)k × (Uk−1\Uk) ⊂ ⋃ k>2(n−1) V(n−1)k × (Uk−1\Uk) ⊂ Wn−1. Taking into account that both the sequences {Un} and {Vn} consist of invariant neighborhoods, we conclude that the sets Wn are invariant as well. Hence the Condition 4 holds too. Now, using the sequence (Wn) we shall produce a sequence (On) sat- isfying all the Pontryagin Conditions 1–5. For every n ∈ ω put On = ⋃∞ i=n WnWn+1 · · ·Wi. Thus Wn ⊃ On+1 ⊃ Wn+1 and On ∩ H × {eT } = {(eH , eT )} for all n. It is easy to see that the sequence {On} consists of invariant sets and satisfies Pontryagin conditions 1–4. Hence the family {On} is a neighborhood base at the unit of some (not necessarily Haus- dorff) topology τ ′ on G = H × T turning G into a paratopological SIN- group. Applying Proposition 1.3 from [Ra1] we conclude that the family Bρ = {OU : O ∈ Bτ ′ , U ∈ Bτ} is a neighborhood base at the unit of some (not necessarily Hausdorff) semigroup topology ρ on G (here we identify H with the subgroup H × {eT } in G). Since the topology ρ is stronger than the product topology π of the group (H, σ) × T [, the topology ρ is Hausdorff and H is a [-closed subgroup of the group (G, ρ). It follows from the construction of the topology ρ that ρ\|H = τ , χ(G, ρ) = χ(H) and \|G/H\| = \|T \|. At the end of the proof we show that the paratopological group (G, ρ) is saturated and [-regular. To show that the group (G, ρ) is saturated it suffices to find for every n ≥ 1 nonempty open sets V ⊂ (H, σ) and U ⊂ T such that V × U−1 ⊂ Wn. Taking into account that the group T is saturated and the set U3n−1\U3n [ is nonempty, find a nonempty open set U ⊂ T such that U−1 ⊂ U3n−1\U3n [ . Then V −1 3n2 × U−1 ⊂ V3n2×(U3n−1\U3n) ⊂ Wn. This implies that the group (G, ρ) is saturated and (G, ρ[) = (H, σ) × T [ ∈ G. The [-regularity of the group (G, ρ) will follow as soon as we prove that WnV π ⊂ Wn−1V for every n ≥ 2 and V ∈ Bτ . Indeed, in this case, Jo u rn al A lg eb ra D is cr et e M at h .16 On subgroups of saturated paratopological groups we shall get On+1V [ ⊂ On+1V π ⊂ WnV π ⊂ Wn−1V ⊂ On−1V. Fix any x ∈ WnV π . If x ∈ V ×{eT }, then x ∈ Wn−1V . Next, assume that x /∈ H×{eT }. The property (4) of the sequence (Uk) implies that the point x has a π-neighborhood meeting only finitely many sets H × Ui, i ∈ ω. This observation together with x ∈ WnV π and (?) imply that x ∈ VniV × (Ui−1 \ Ui) [ for some i > 2n. The condition (6) implies that the following chain of inclusions holds: x ∈ VniV × (Ui−1 \ Ui) [ ⊂ VniV σ × (Ui−1 \ Ui) [ ⊂ V 2 niV × (Ui−2 \ Ui+2) ⊂ i+2 ⋃ j=i−1 Vni−1V × (Uj−1 \ Uj) ⊂ ⋃ j>2n−2 V(n−1)jV × (Uj−1 \ Uj) ⊂ Wn−1V. Finally, assume that x ∈ H \ V = (H \ V ) × {eT }. Since the set V is [-closed in H, there is m ∈ ω such that V −1 m Vmx ∩ V = ∅ and thus Vmx ∩ ViV = ∅ for all i ≥ m. The inclusion x ∈ WnV π and (?) imply (Vm × UmU−1 m )x ∩ (VniV × (Ui−1 \ Ui)) 6= ∅ for some i > 2n. Then Vmx ∩ VniV 6= ∅ and UmU−1 m ∩ (Ui−1 \ Ui) 6= ∅. In view of Property (5) of the sequence (Uk), the latter relation implies m ≤ i. On the other hand, the former relation together with the choice of the number m yields ni < m ≤ i which is impossible. This contradiction finishes the proof of the inclusion WnV π ⊂ Wn−1V . 3. Proof of Theorem 3 Given a topological space (X, τ) Stone [Sto] and Katetov [Kat] considered the topology τr on X generated by the base consisting of all canonically open sets of the space (X, τ). This topology is called the regularization of the topology τ . If (X, τ) is Hausdorff then (X, τr) is regular and if (X, τ) is a paratopological group then (X, τr) is a paratopological group too [Ra2, Ex.1.9]. If (G, τ) is a paratopological group then τr is the strongest regular semigroup topology on the group G which is weaker than τ ; moreover, for any neighborhood base B at the unit of the group (G, τ) the family Br = {intU : U ∈ B} is a base at the unit of the group (G, τr) [Ra3, p.31–32]. The following proposition is quite easy and probably is known. Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 17 Proposition 6. Let (X, τ) be a topological space. Then (X, τ) is pseu- docompact if and only if the regularization (X, τr) is pseudocompact. For the proof of Theorem 3 we shall need a special pseudocompact functionally Hausdorff semigroup topology on the unit circle. We recall that a topological space X is functionally Hausdorff if continuous func- tions separate points of X. Proposition 7. There is a functionally Hausdorff pseudocompact first countable semigroup topology θ on the unit circle T which is not a group topology. Proof. Let T be the unit circle and χ : T → Q be a (discontinuous) group homomorphism onto the groups of rational numbers. Fix any element x0 ∈ T with χ(x0) = 1 and observe that S = {1} ∪ {x ∈ T : χ(x) > 0} is a subsemigroup of T. Let θ be the weakest semigroup topology on T containing the standard compact topology τ and such that S is open in θ. It is easy to see that θ is functionally Hausdorff and the sets S ∩ U , where 1 ∈ U ∈ τ , form a neighborhood base of the topology θ at the unit of T. By Proposition 6, to show that the group (T, θ) is pseudocompact it suffices to verify that θr = τ . Since τ is a regular semigroup topology on the group T weaker than θ, we get θr ⊃ τ . To verify the inverse inclusion we first show that U τ = U θ for any U ∈ θ. Since τ ⊂ θ it suffices to show that U τ ⊂ U θ . Fix any point x ∈ U τ and a neighborhood V ∈ τ of 1. We have to show that x(V ∩ S) ∩ U 6= ∅. Pick up any point y ∈ xV ∩ U . Since U is open in the topology θ, we can find a neighborhood W ∈ τ of 1 such that y(W ∩S) ⊂ xV ∩U . Find a number N such that χ(yxN 0 ) > χ(x) and thus yxn 0 ∈ xS for all n ≥ N (we recall that x0 is an element of T with χ(x0) = 1). Moreover, since x0 is non- periodic in T, there exists a number n ≥ N such that xn 0 ⊂ W . Then yxn 0 ∈ (yS ∩ yW ) ∩ xS ⊂ (xV ∩ U) ∩ xS = x(V ∩ S) ∩ U . Hence x ∈ U θ and U θ = U τ . Then intθ U θ = T\T\U θ θ = T\T\U θ τ ∈ τ which just yields θr ⊂ τ . Now we are able to present a proof of Theorem 3. The “if” part follows from the observation that for any Hausdorff pseudocompact paratopolog- ical group (G, τ) its group reflexion G[ = (G, τr) is a Hausdorff pseudo- compact (and hence totally bounded) topological group [RR]. Jo u rn al A lg eb ra D is cr et e M at h .18 On subgroups of saturated paratopological groups To prove the “only if” part, fix a Bohr-separated abelian paratopolog- ical group (H, τ) and let Bτ be a neighborhood base at the unit of the group (H, τ). It follows that there is a group topology σ′ ⊂ τ on H such that (H, σ′) is totally bounded. Let (Ĥ, σ) be the Raikov completion of the group (H, σ′). It is clear that Ĥ is a compact abelian group and H is a normal dense subgroup of Ĥ. It follows that Bτ is a neighborhood base at the unit of some semigroup topology τ ′ on the group Ĥ with τ ′\|H = τ . Let (T, θ) be the group from Proposition 7. We shall define the topology on the product G = Ĥ × T as follows. Let Bτ , Bσ and Bθ be the open neighborhood bases at the unit of the groups (H, τ), (Ĥ, σ) and (T, θ) respectively. For arbitrary neighborhoods Uτ ∈ Bτ , Uσ ∈ Bσ and Uθ ∈ Bθ with Uτ ⊂ Uσ let [Uτ , Uσ, Uθ] = Uτ × {eT} ∪ Uσ × (Uθ\{eT}), where eH and eT are the units of the groups H and T respectively. Denote by B the family of all such [Uτ , Uσ, Uθ]. Repeating the argument of Theorem 1 check that the family B is a base of some Hausdorff semigroup topology ρ on G. By π denote the topology of the product (Ĥ, σ)× (T, θr). By Proposition 6 to show that the group (G, ρ) is pseudocompact it suffice to verify that ρr ⊂ π. For this we shall show that U ρ ⊃ Uσ × U θ θ for every U = [Uτ , Uσ, Uθ] ∈ B. Let (xĤ , xT) ∈ Uσ × U θ θ and V = [Vτ , Vσ, Vθ] ∈ B. It suffice to show that ( (xĤ , xT) + Vσ × (Vθ\{eT}) ) ∩ Uσ × (Uθ\{eT}) 6= ∅. This intersection is nonempty if and only if the intersections (xĤ + Vσ) ∩ Uσ and (xT + (Vθ\{eT}))∩ (Uθ\{eT}) are nonempty. The first intersection is nonempty since xĤ ∈ Uσ and the second is nonempty since xT ∈ U θ θ and the topology θ is non-discrete. References [AH] A.V. Arhangel’skii, M. Hušek, Extensions of topological and semitopological groups and the product operation // Comment. Math. Univ. Carolinae. 42:1 (2001) 173-186. [A] L. Außenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups // Dissert. Math. 384 (1999). [Ban] T. Banakh, On topological groups containing a Fréchet-Urysohn fan // Matem. Studii. 9:2 (1998), 149–154. [Ba] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lect. Notes in Math., Vol.1466, Springer, Berlin, 1991. [BR1] T. Banakh, O. Ravsky. Oscillator topologies on a paratopological group and related number invariants, // Algebraical Structures and their Applications, Kyiv: Inst. Mat. NANU, (2002) 140-153. [BR2] T. Banakh, O. Ravsky. Asymmetry indices of paratopological groups, preprint. [BR3] T. Banakh, O. Ravsky. On completeness properties of paratopological groups, (in preparation). Jo u rn al A lg eb ra D is cr et e M at h .T. Banakh, S. Ravsky 19 [BRZ] T. Banakh, S.Ravsky, L. Zdomsky. On the character of the group topology generated by a countable T -filter, (in preparation). [BG] B. Bokalo, I. Guran. Sequentially compact Hausdorff cancellative semigroup is a topological group // Matem. Studii 6 (1996), 39–40. [Cho] M.M. Choban. Some topics in topological algebra // Topology Appl. 45 (1993), 183–202. [Ed] R.E.Edwards. Functional Analysis. Theory and Applications, NY: Holt, Rine- hart and Winston, 1965. [FT] S.P. Franklin, B.V. Smith Thomas. A survey of kω-spaces // Topology Proc. 2 (1977), 111–124. [Gra] M.I. Graev. Free topological groups // Izvestia AN SSSR. Ser. Mat. 12:3 (1948), 279–324. [Gu] I.Y. Guran. Cardinal invariants of paratopological groups // in: II Intern. Algebraic Conf. in Ukraine, – Kyiv - Vinnytsia, 1999 (in Ukrainian). [JW] W. Just, M. Weese. 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Dedicated to the 110th anniversary of Stefan Banach (May 28-31, 2002) – Lviv, 2002. – P.170–172. [Sto] M.H. Stone. Applications of the theory of Boolean rings to general topology // Trans. Amer. Math. Soc. 41 (1937), 375–481. [Tk] M. Tkachenko. Introduction to topological groups // Topology Appl. 86 (1998), 179–231. [Va] J.E. Vaughan. Small uncountable cardinals and topology // in: Open Prob- lems in Topology (J. van Mill and G.M.Reed eds.), Amsterdam: North- Holland, 1990. – P.195–216. Jo u rn al A lg eb ra D is cr et e M at h .20 On subgroups of saturated paratopological groups Contact information T. Banakh Instytut Matematyki, Akademia Świȩtokrzyska in Kielce, Świȩtokrzyska 15, Kielce, 25406, Poland and Department of Mathematics, Ivan Franko Lviv National University, Universytetska, 1 Lviv 79000, Ukraine E-Mail: tbanakh@franko.lviv.ua S. Ravsky Department of Mathematics, Ivan Franko Lviv National University, Universytetska, 1 Lviv 79000, Ukraine E-Mail: oravsky@mail.ru

On subgroups of saturated or totally bounded paratopological groups

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