Semigroups of Sets Without the Baire Property In Finite Dimensional Euclidean Spaces Vénuste NYAGAHAKWA

Linköping Studies in Science and Technology. Thesis No. 1700 Semigroups of Sets Without the Baire Property In Finite Dimensional Euclidean Spaces Vénuste NYAGAHAKWA Department of Mathematics Division of Mathematics and Applied Mathematics Linköping University, SE 581 83 Linköping, Sweden Linköping 2015

Linköping Studies in Science and Technology. Thesis No. 1700 Semigroups of Sets Without the Baire Property In Finite Dimensional Euclidean Spaces Vénuste NYAGAHAKWA venuste.nyagahakwa@liu.se www.mai.liu.se Mathematics and Applied Mathematics Department of Mathematics Linköping University SE 581 83 Linköping Sweden ISBN 978-91-7519-146-1 ISSN 0280-7971 Copyright 2015 Vénuste NYAGAHAKWA Printed by LiU-Tryck, Linköping, Sweden 2015

To Iribagiza M. Rose and Mizero B. Roberto.

Abstract A semigroup of sets is a family of sets closed under finite unions. This thesis focuses on the search of semigroups of sets in finite dimensional Euclidean spaces R n, n 1, which elements do not possess the Baire property, and on the study of their properties. Recall that the family of sets having the Baire property in the real line R, is a σ algebra of sets, which includes both meager and open subsets of R. However, there are subsets of R which do not belong to the algebra. For example, each classical Vitali set on R does not have the Baire property. It has been shown by Chatyrko that the family of all finite unions of Vitali sets on the real line, as well as its natural extensions by the collection of meager sets, are (invariant under translations of R) semigroups of sets which elements do not possess the Baire property. Using analogues of Vitali sets, when the group Q of rationals in the Vitali construction is replaced by any countable dense subgroup Q of reals, (we call the sets Vitali Q-selectors of R) and Chatyrko s method, we produce semigroups of sets on R related to Q, which consist of sets without the Baire property and which are invariant under translations of R. Furthermore, we study the relationship in the sense of inclusion between the semigroups related to different Q. From here, we define a supersemigroup of sets based on all Vitali selectors of R. The defined supersemigroup also consists of sets without the Baire property and is invariant under translations of R. Then we extend and generalize the results from the real line to the finite-dimensional Euclidean spaces R n, n 2, and indicate the difference between the cases n = 1 and n 2. Additionally, we show how the covering dimension can be used in defining diverse semigroups of sets without the Baire property. v

Populärvetenskaplig sammanfattning En semigrupp av mängder är en familj av mängder som är sluten under ändliga unioner. Denna licentiatavhandling fokuserar på sökandet efter semigrupper av mängder i de ändligdimensionella euklidiska rummen R n, n 1, vars element inte har Baire-egenskapen, och på studiet av deras egenskaper. Som bekant är familjen av delmängder av R som har Baire-egenskapen en σ- algebra av mängder som innehåller både de öppna och de magra delmängderna av R. Det finns dock delmängder av R som inte tillhör algebran. Till exempel gäller att varje klassisk Vitali-mängd på R inte har Baire-egenskapen. Chatyrko har visat att familjen av alla ändliga unioner av Vitali-mängder på reella linjen, och den utvidgning av denna familj som på ett naturligt sätt ges av familjen av magra mängder, är semigrupper av mängder på R, som består av mängder som inte har Baire-egenskapen och som är invarianta under translationer av R. Genom att använda motsvarigheten till Vitali-mängder när gruppen Q i Vitalis konstruktion ersätts med en godtycklig uppräknelig tät delgrupp Q av R (vi kallar dessa mängder Vitali-Q-selektorer på R), och Chatyrkos metoder, konstruerar vi semigrupper av mängder på R relaterade till Q, som består av mängder som inte har Baire-egenskapen och som är invarianta under translationer av R. Vidare studerar vi hur semigrupper relaterade till olika grupper Q förhåller sig med avseende på inklusion, och vi definierar en supersemigrupp baserad på alla Vitali-selektorer på R. Denna supersemigrupp består också av mängder som inte har Baire-egenskapen och är invariant under translationer av R. Vi utvidgar och generaliserar sedan dessa resultat från reella linjen till ändligdimensionella euklidiska rum R n, n 2, och visar på olikheterna i de två fallen n = 1 och n 2. Dessutom visar vi hur begreppet övertäckningsdimension kan användas för att definiera en mångfald av semigrupper av mängder som inte har Baireegenskapen. vii

Acknowledgments This Licentiate thesis appears in its current form due to the assistance and guidance of several people. I would therefore like to offer my sincere thanks to all of them. First, I would like to express my gratitude to my supervisors Vitalij A. Chatyrko and Mats Aigner for useful comments, remarks and engagement through the learning process until to this Licentiate thesis. They provided encouragement and advise necessary for me to complete this Licentiate thesis and to proceed through the Ph.D program. In a very special way, I would also like to thank Bengt Ove Turesson, Björn Textorius and all members of the Department of Mathematics for their helps whenever need arises at work. They have directed me through various situations, allowing me to reach this accomplishment. I thank Minani Froduald, Lyambabaje Alexandre and Mahara Isidore for their assistance and guidance and the helpful part they played in my mathematical developments as my teachers. I would like to thank all my fellow Ph.D students whom I shared so many excellent times at Linköping University. In particular, I would like to thank Anna Orlof for organizing team building activities. My family has supported and helped me along the learning process by giving encouragement and providing the moral and emotional support I needed to complete my thesis. To them, I am eternally grateful. I wish to acknowledge the financial support I received through Sida/Sarec funded University of Rwanda-Linköping University cooperation. All involved institutions and people are hereby acknowledged. May the Almighty God richly bless all of you. Linköping, March 24, 2015 Vénuste NYAGAHAKWA ix

Contents 1 Introduction 3 1.1 Problem formulation........................... 3 1.2 Summary of main results........................ 4 1.3 Structure of the thesis.......................... 6 2 Necessary facts 9 2.1 Algebraic notions in set theory..................... 9 2.2 Some topological concepts........................ 11 2.2.1 Baire Category Theorem..................... 11 2.2.2 Baire property........................... 13 2.2.3 Lebesgue covering dimension.................. 13 3 Algebra of semigroups of sets 15 3.1 Semigroups and ideals of sets...................... 15 3.2 Extension of a semigroup of sets via an ideal of sets......... 18 4 Semigroups of sets defined by Vitali selectors on the real line 23 4.1 Vitali selectors of the real line...................... 24 4.2 Countable dense subgroups of R and generated semigroups.... 28 4.3 Supersemigroup based on Vitali selectors of R............ 31 xi

xii Contents 4.4 Semigroup of non-lebesgue measurable sets............. 34 5 Semigroups of sets without the Baire property in finite dimensional Euclidean spaces 35 5.1 Vitali selectors of R n........................... 35 5.2 Supersemigroup of Vitali selectors of R n............... 39 5.3 Rectangular Vitali selectors of R n.................... 41 5.3.1 Supersemigroup of rectangular Vitali selectors of R n.... 46 5.4 Semigroups of sets in R n defined by dimension........... 49 Bibliography 55

Notation The principal notation used throughout the text is listed below. N Z Q R R n Int X A Cl X A Y c I f I c I cd I n M M n N 0 The set of positive integers The set of integers The set of rational numbers The set of real numbers The n dimensional Euclidean space Interior of a set A in a topological space X Closure of a set A in a topological space X Complement of a set Y Ideal of finite sets Ideal of countable sets Ideal of closed and discrete sets Ideal of nowhere dense sets σ ideal of meager sets in R σ ideal of meager sets in R n The family of all subsets of R having Lebesgue measure zero 1

2 Notation N The family of all measurable subsets of R in the Lebesgue sense,,, \ Standard set operation of symmetric difference, union, intersection and set difference dim Lebesgue covering dimension P(X) The family of all subsets of X O The family of all open subsets of R O n B p B n p F The family of all open subsets of R n The family of sets with the Baire property on the real line The family of sets with the Baire property in R n The family of all countable, dense in the real line subgroups of (R, +) F n The family of all countable, dense in the Euclidean spaces R n, n 2 subgroups of (R n, +) rf n The family of all rectangular subgroups of (R n, +) S A I A V(Q) S V(Q) S V sup V n (Q) S V n Q rv n (Q) S rv n Q Semigroup of sets generated by A Ideal of sets generated by A Family of all Vitali-Q selectors of R associated to the subgroup Q of (R, +) Semigroup of sets generated by V(Q) Semigroup of sets generated by V sup The family of all Vitali-Q selectors of R n associated to the subgroup Q of (R n, +) Semigroup of sets generated by V n (Q) The family of all rectangular Vitali-Q selectors of R associated to the subgroup Q of (R n, +) Semigroup of sets generated by rv n (Q) S rv n sup Semigroups of sets generated by rv n sup

1 Introduction 1.1 Problem formulation Let R be the set of real numbers and P(R) the family of all subsets of R. Furthermore, let (R, τ E ) be the real line, i.e. the set R endowed with the topology τ E defined by all open intervals of R, and M the family of all meager subsets of (R, τ E ). An interesting extension of M, as well as τ E in P(R), is the family B p of all subsets of (R, τ E ) possessing the Baire property. Recall ([1]) that a set B B p if and only if there are an O τ E and an M M such that B = O M. It is well known (see [1]) that B p = P(R) (for example, each Vitali set V of R [2] is an element of the complement B c p = P(R) \ B p of B p in P(R)), and the family B p is a σ algebra of sets, in particular, B p is closed under finite unions and finite intersections of sets. Let us also note that the family B p is invariant under action of the group H((R, τ E )) of all homeomorphisms of the real line (R, τ E ) onto itself, i.e. for each B B p and each h H((R, τ E )) we have h(b) B p. It is easy to see that the family B c p is also invariant under action of the group H((R, τ E )) but, unlike the family B p, B c p is not closed under finite unions and finite intersections of sets. It is also well known (cf. [3]) that there are elements of B c p with a natural algebraic structure (for example, some subgroups of the 3

4 1 Introduction additive group (R, +) of all real numbers). One can pose the following problem ([4]): Do there exist subfamilies of B c p which are invariant under action of an infinite subgroup of H((R, τ E )) and on which we can define some algebraic structure? The following simple observation can give an answer to the question. Let G be a non-trivial sufficiently rich subgroup G of H((R, τ E )) and A be a subfamily of B c p such that A is invariant under action of G. If for each n 2 and each A 1, A 2,, A n A we have A 1 A 2 A n B c p then the family S A consisting of all finite unions of elements of A is a semigroup of sets with respect to the binary operation union of sets, which is invariant under action of G and which is in B c p. In [5], Chatyrko proved that any union of finitely many Vitali sets is an element of B c p. It is easy to see that the family V of all Vitali sets is invariant under action of the group τ(r) of all translations of R. Hence, by the observation above, the family S V of all finite unions of Vitali sets is a semigroup of sets with respect to the operation " ", which is invariant under action of τ(r) and which is in B c p. Furthermore, in [4] Chatyrko proved that the family S V M = {U M : U S V, M M} is also a semigroup of sets with the respect to the operation " ", which is invariant under action of τ(r) and which is in B c p. (a) The goals of this thesis are the following. In the realm of P(R) to find families of sets (different from the families S V and S V M mentioned above) which are semigroups of sets with respect to the operation, which are invariant under action of τ(r) and which are in B c p. (b) To extend the results of (a) to Euclidean spaces R n, n 2. 1.2 Summary of main results Let X be a non-empty set and let P(X) be the family of all subsets of X. For families of sets A and B in P(X), we define two operations: A B = {A B : A A, B B}, A B = {A B : A A, B B}. However, by A B we denote the family of common elements of A and B.

1.2 Summary of main results 5 Moreover, if D P(X) then by S D we mean the family of all finite unions of elements of D. In Chapter 3, we have obtained the following results (Proposition 3.2 and Proposition 3.4): (a) If S is a semigroup of sets with respect to the operation and I is an ideal of sets in P(X), then the families S I, S I are semigroups of sets with respect to the operation and S S I S I. (b) Let I be an ideal of sets and A, B P(X) such that A I = and for each element U S A and each non-empty element B B there is an element A A satisfying A B \ U. Then (S A I) (S B I) =. In Chapter 4, the main attention was paid to the case when X = R. Let Q be a countable, dense in the real line, subgroup of (R, +), V(Q) be the family of all Vitali Q-selectors of R associated to Q (analogues of Vitali sets, considered in [3], when the set Q of rationals is substituted by Q) and I any subideal of M. Using the Chatyrko s method, the results from the previous chapter and the observation that τ E = S τe, we have proved (Proposition 4.3) that: (c) The families S V(Q), S V(Q) I and S V(Q) I are semigroups of sets with respect to the operation and S V(Q) S V(Q) I S V(Q) I. Moreover, S V(Q), S V(Q) I and S V(Q) I are invariant under action of τ(r), and consist of sets without the Baire property. We have also observed that in the family {S V(Q) : Q R} there is no element which contains all others (Proposition 4.8). So we consider the family V sup of all Vitali Q-selectors of R, where Q is varied, and the semigroup S V sup which we call a supersemigroup of Vitali selectors. The supersemigroup S V sup contains the semigroup S V(Q) for each Q. In the same way as above we have proved (Theorem 4.1) that: (d) The families S V sup, S V sup I and S V sup I are semigroups of sets with respect to the operation and S V sup S V sup I S V sup I. Moreover, S V sup, S V sup I and S V sup I are invariant under action of τ(r), and consist of sets without the Baire property.

6 1 Introduction In Chapter 5, the main attention was paid to the case when X = R n, n 2. Let Q be a countable, dense in the Euclidean space R n, subgroup of (R n, +), V n (Q) be the family of all Vitali Q-selectors of R n associated to Q (analogues of Vitali Q-selectors for the real line, see also [3]) and I any subideal of the family M n of all meager sets of R n. Using the results and technique from the previous chapters we have proved statements which are similar to (c) and (d). Moreover, we have pointed out a special case of Vitali Q-selectors of R n called rectangular Vitali selectors of R n, n 2, related to groups Q which are products of n many countable dense in the real line groups. Our rectangular Vitali selectors are supposed to be products of n many Vitali selectors of the real line. We have extended our theory to the special case. Since the rectangular products are somewhat less complicated than general ones, we could obtain more information about them. A part of the mentioned results can be found in [4] and [6]. 1.3 Structure of the thesis This thesis contains five chapters which are structured as follows: The first chapter gives a brief description of the problem under investigation and a brief summary of the obtained results. The second chapter introduces basic concepts, terminology and statements which will be needed for a better understanding of the subsequent chapters. The third chapter treats the theory of semigroups of sets with respect to the operation of union of sets. Through various examples, we describe the behaviour of semigroups of sets with the respect to several binary operations. Furthermore, we present a way of searching pairs of semigroups of sets without common elements. The fourth chapter develops the theory of semigroups of sets without the Baire property on the real line. These semigroups are constructed by using Vitali Q-selectors of R and subideals of the ideal of meager subsets of R. They are invariant under translations of the real line and they consist of sets without the Baire property. The last fifth chapter generalizes and extends the results of Chapter 4 to

1.3 Structure of the thesis 7 the finite-dimensional Euclidean spaces R n, n 2. Besides the semigroups of sets generated by ordinary Vitali selectors, the chapter treats also semigroups of sets generated by rectangular Vitali selectors of R n, n 2. Rectangular Vitali selectors are a special case of the ordinary ones. In both cases, the generated semigroups of sets are invariant under translations of R n and they consist of sets without the Baire property. The chapter ends by pointing out the role of dimension in defining different semigroups of sets without the Baire property.

2 Necessary facts The purpose of this chapter is to recall concepts and terminology which will be used in the subsequent chapters. 2.1 Algebraic notions in set theory In this section, we shall give a short introduction to families of sets with algebraic properties. By a family of sets, we mean any set whose elements are themselves sets. Families of sets are denoted by capital script letters like S, O and so forth. For a more detailed information, we refer the reader to one of the references [7], [8] or [9]. Let X be a non-empty set and let P(X) be the family of all subsets of X. Definition 2.1. A non-empty family R P(X) of sets is called a ring of sets on X if A B R and A B R whenever A R, B R. Since A B = (A B) (A B), A \ B = A (A B), we have also A B R and A \ B R whenever A R, B R. Thus a ring is a family of sets closed under the operations of taking unions, intersections, differences and symmetric differences. A ring of sets must contain the empty set, since A \ A =. Definition 2.2. Let A P(X) be a ring of sets on X. If X A, the family A is called an algebra of sets on X. 9

10 2 Necessary facts From this definition, a ring of sets is an algebra if and only if it closed under taking the operation of complement. Example 2.1 (i) The family of all finite subsets of X is a ring on X but not an algebra on X unless X is finite. (ii) Let R be the set of real numbers. The family of all bounded subsets of R is a ring on R but not an algebra. Definition 2.3. (a) A ring R P(X) is called a σ-ring of sets on X if it is closed under countable unions, i.e. it contains the union S = n=1 A n whenever it contains the sets A 1, A 2,.... (b) A σ-ring A P(X) is called a σ-algebra of sets on X if X A. From the De Morgan formula n=1 A n = X \ n=1 (X \ A n ), it follows that each σ-algebra is also closed under countable intersection of sets. Note that that a σ algebra can be defined as an algebra closed under countable unions. Example 2.2 For a set X, the family of all countable subsets of X is a σ ring. It will be a σ algebra if X is countable. Definition 2.4. Let A P(X). The smallest σ algebra of sets on X containing A is called a σ algebra generated by the family A. Example 2.3 Let R be the real line, i.e. the set R endowed with the standard metric ρ defined by ρ(x, y) = x y for all x, y R and N be the family of all measurable subsets of R in the Lebesgue sense. The family N is a σ algebra of sets which is generated by N 0 O, where N 0 is the family of all subsets of R having Lebesgue measure zero and O is the family of all open subsets of R.

2.2 Some topological concepts 11 Definition 2.5. (a) A family I P(X) of sets is called an ideal of sets on X, if it satisfies the following two conditions: (i) If A I and B I then A B I. (ii) If A I and B A then B I. (b) If an ideal of sets I is closed under countable unions, then it is called a σ ideal of sets on X. Example 2.4 Let A X. Then (i) the family I(A) = {B : B A} is a σ ideal of sets on X, (ii) the family I c of all countable subsets of X forms a σ ideal of sets on X, (iii) the family I f of all finite subsets of X forms an ideal of sets on X, but not a σ ideal of sets, whenever X is infinite. 2.2 Some topological concepts 2.2.1 Baire Category Theorem Let X be a topological space and let A be a subset of X. Recall that a neighborhood of a point x X is any open subset U of X containing x. The point x X is a limit point of A if (U \ {x}) A = for every neighborhood U of x. The derived set of A, denoted by A d, is the set of all limit points of A. Definition 2.6. A subset A of X is said to be closed and discrete if and only if A d =. Note that each subset of a closed and discrete subset of X (resp. each finite union of closed discrete subsets of X) is also a closed discrete subset of X. Thus the family of all closed and discrete subsets of X forms an ideal of sets, denoted by I cd.

12 2 Necessary facts Definition 2.7. A subset A X is called a nowhere dense set in X if Int X (Cl X (A)) =. It is easy to see that every subset of a nowhere dense set is a nowhere dense set, and the union of finitely many nowhere dense sets is again a nowhere dense set. Thus, the family of nowhere dense sets in a given topological space forms an ideal of sets, denoted by I n. Example 2.5 Every finite subset of the real line R, the set Z of all integers and the Cantor set, are nowhere dense subsets of R. Definition 2.8. A subset A X is said to be dense in X if Cl X (A) = X. Example 2.6 (1) The set Q of all rational numbers is a dense subset of R. (2) The set Z( 2) = {n + 2m : n Z, m Z} is a dense subset of R [10]. Remark 2.1. Note that a countable union of nowhere dense sets is not necessarily a nowhere dense set. For example, the set Q of rationals is a union of countably many nowhere dense sets in R, but Int R (Cl R (Q)) = R. Definition 2.9. A subset A X is meager (or of first category) if A is the union of countably many nowhere dense sets. Any set that is not meager is said to be nonmeager (or of second category). Example 2.7 The set Q of rationals numbers is a meager subset of the real line R. In a given topological space, the family of all meager sets forms a σ ideal of sets. The σ ideal of meager sets will be denoted by M in our further considerations. A fundamental theorem of Baire asserts the following [11], [12].

2.2 Some topological concepts 13 Theorem 2.1 (Baire Category Theorem). Let X be a complete metric space. Then X can not be covered by countably many nowhere dense subsets. Moreover, the union of countably many nowhere dense subsets of X has a dense complement. Recall that the real line R is a complete metric space. So by Baire Category Theorem, R is of the second category. Similarly, the Euclidean space R n, n 1, i.e. the set R n endowed with the metric ρ(x, y) = i=1 n (x i y i ) 2, where x = (x 1,, x n ) and y = (y 1,, y n ), is a complete metric space. So R n is of the second category. Remark 2.2. On the real line R the ideals of sets I f, I cd, I c, I n and M satisfy the inclusions I f I cd I c M and I f I cd I n M. Note that I c and I n are not comparable in the sense of inclusion on R. In fact, the Cantor set is uncountable and nowhere dense subset of R while the set Q of rationals numbers is a countable dense subset of R. 2.2.2 Baire property In this subsection, X is assumed to be a topological space. Definition 2.10. A subset A X is said to have the Baire property if it can be represented in the form A = O M, where O is an open set of X and M is a meager set of X. (Recall that O M = (O \ M) (M \ O)) Note that a subset A X has the Baire property in X if and only if there is an open set O of X and two meager sets M, N of X such that A = (O \ M) N. The family of all sets with the Baire property in a topological space X will be denoted by B p in our further considerations. Recall that the family B p is a σ-algebra of sets. In particular, each open set of X and each meager set of X have the Baire property. Thus, the σ-algebra B p is the smallest σ-algebra containing all open and all meager sets in X. 2.2.3 Lebesgue covering dimension In this subsection, we present some basic properties of Lebesgue covering dimension dim. Let X be a topological space and let A = {A α } α Γ be a family of subsets of X, where Γ is an index set.

14 2 Necessary facts Definition 2.11. (i) The order of the family A = {A α } α Γ of subsets, not all empty, of X, is the largest integer n for which there exists a subset I of Γ with n + 1 elements such that α I A α is non-empty, or if there is no such largest integer. (ii) The family A = {A α } α Γ is a cover of X if α Γ A α = X. (iii) A cover B is a refinement of another cover A of the same space X, if for every B B there exists A A such that B A. Definition 2.12. Let X be a topological space. Then dim X = 1 if and only if X =. dim X n if each finite open cover of X has an open refinement of order not exceeding n. dim X = n if it is true that dim X n but it is not true that dim X n 1. dim X = if for every integer n it is false that dim X n. If dim X = n, then X is called the n dimensional topological space. Recall that a space X is said to be separable if it contains a countable dense subset. It is said to be metrizable if there exists a metric on X which induces the topology on X. For separable metrizable spaces, some basic properties about Lebesgue covering dimension are summarized in the following theorems [13]. Theorem 2.2 (Fundamental Theorem of Dimension). For every natural number n, we have dim R n = n. Theorem 2.3 (Monotonicity). If A is a subspace of a separable metrizable space X, then dim A dim X. Theorem 2.4 (Countable Sum Theorem). Let X be a separable metrizable space and X = i=1 F i where F i is closed in X for each i. If dim F i n for each i, then dim X n. Theorem 2.5 (Product Theorem). Let X and Y be non-empty separable metrizable spaces. Then dim(x Y) dim X + dim Y. Theorem 2.6 (Brouwer Dimension Theorem). Let X R n. Then dim X = n if and only if Int R n(x) =.

3 Algebra of semigroups of sets In this chapter we introduce the notion of a semigroup of sets. Then we look at the behaviour of semigroups of sets under some binary operations. Additionally, we present a way to extend a given semigroup of sets to another one by the use of ideals of sets. Before ending the chapter, we state and prove a proposition which will be used in searching of pairs of semigroups of sets without common elements. The results of this chapter were taken from the article [6]. Below X is assumed to be a non-empty set and P(X) is the family of all subsets of X. 3.1 Semigroups and ideals of sets Families of sets, like rings of sets or algebra of sets, are of fundamental importance in Topology and Analysis, and their properties are well known (see [9], [14]). In this section, we will consider another families of sets, namely, semigroups of sets and ideal of sets, and prove some statements about them. Definition 3.1. A non-empty set S is called a semigroup if there is a binary operation : S S S for which the associativity law is satisfied, i.e. the equality (x y) z = x (y z) holds for all x, y, z S. The semigroup S is called abelian if x y = y x for all x, y S. 15

16 3 Algebra of semigroups of sets Consider a family of sets S P(X) such that for each pair of elements A, B S we have A B S. Since the union of sets is both commutative and associative, such a family of sets will be an abelian semigroup with the respect to the operation of union of sets. This observation leads to the following definition. Definition 3.2. A non-empty family of sets S P(X) is called a semigroup of sets on X if it is closed under finite unions. Remark 3.1. Using the definition of a semigroup of sets, we can redefine the notion of an ideal of sets in the following way: a non-empty family I P(X) is an ideal of sets on X iff it is a semigroup of sets on X and if A I and B A then B I. Let A P(X). Put S A = { n i=1 A i : A i A, n N} and I A = {B P(X) : there is A S A such that B A}. The following proposition is evident. Proposition 3.1. The family S A is a semigroup of sets on X and the family I A is an ideal of sets on X. We will call S A the semigroup of sets generated by the family A and I A will be called the ideal of sets generated by the family A. Let us define three binary operations on subfamilies of P(X) as follows. If A, B P(X) then (1) A B = {A B : A A, B B}; (2) A B = {A B : A A, B B}; (3) A B = {(A \ B 1 ) B 2 : A A; B 1, B 2 B} where, and \ are the usual union, symmetric difference of sets and difference of sets, respectively. For the defined operations, we observe the following: (i) Since the union and the symmetric difference of sets are commutative operations, we have A B = B A and A B = B A. From the fact that A B = (A \ B) B = (B \ A) A, we have A B A B and A B B A. We note that if A and B are both semigroups of sets or ideals of sets, then the family A B is of the same type.

3.1 Semigroups and ideals of sets 17 (ii) As we will see in the following examples, in general for given semigroups of sets A and B, the families A B, A B, B A do not need to be semigroups of sets and none of the inclusions A B A B, A B A B, A B A B, A B A B, A B B A needs to hold. Moreover, one of the families A B, B A can be a semigroup of sets while the other is not. Example 3.1 Let X 2 and A be a non empty proper subset of X. Put B = X \ A, A = {A, X} and B = {B, X}. Note that A = S A, B = S B and the families A B = {X}, A B = {, A, B, X}, A B = {B, X}, B A = {A, X} are semigroups of sets. Moreover, none of the following inclusions A B A B, A B A B, A B B A and B A A B holds. In our further considerations, the notation Y c means the complement of a set Y in the set X. Example 3.2 Let X = {1, 2, 3, 4}, A 1 = {1, 3}, A 2 = {2, 4}, B 1 = {1, 2}, B 2 = {3, 4}, C = {1, 4}, D = {2, 3}, A = {, A 1, A 2 } and B = {, B 1, B 2 }. Note that S A = {, A 1, A 2, X} and S B = {, B 1, B 2, X}. Moreover, we have S A S B = {, A 1, A 2, B 1, B 2, {1} c, {2} c, {3} c, {4} c, X}, S A S B = {, A 1, A 2, B 1, B 2, C, D, X} and S A S B = S B S A = P(X) \ {C, D}. It is easy to see that the inclusions S A S B S A S B and S A S B S A S B do not hold. We note also that none of the families S A S B, S A S B and S B S A are semigroups of sets. In fact, A 1, D S A S B but A 1 D = {4} c / S A S B, and {1}, {4} S A S B but {1} {4} = C / S A S B. Example 3.3 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A 1 = {1, 2, 4, 5, 7, 8}, A 2 = {2, 3, 5, 6, 8, 9}, B 1 = {1, 2, 3, 4, 5, 6}, B 2 = {4, 5, 6, 7, 8, 9}, A = {A 1, A 2 }, B = {, B 1, B 2 }. Note that S A = {A 1, A 2, X} and S B = {, B 1, B 2, X}. First we will show that the family S A S B is not a semigroup of sets. It is enough to prove that the set C = ((A 1 \ B 1 ) ) ((A 2 \ B 2 ) ) / S A S B. It is clear that the set C is given by C = (A 1 \ B 1 ) (A 2 \ B 2 ) = {2, 3, 7, 8}. Assume that C S A S B.

18 3 Algebra of semigroups of sets Thus C = ((S 1 \ S 2 ) S 3 ) for some S 1 S A and S 2, S 3 S B. Since C = 4, we have S 3 =. Let S 1 = A 1. Then S 1 \ S 2 is either 2 (if S 2 is B 1 or B 2 ), 0 (if S 2 = X) or 6 (if S 2 = ). We have a contradiction. If S 1 = A 2, we also have a contradiction by a similar argument as above. Assume now that S 1 = X. Then S 1 \ S 2 is either 3 (if S 2 is B 1 or B 2 ), 0 (if S 2 = X) or 9 (if S 2 = ). We have again a contradiction that proves the statement. Further note that S B S A = {A 1, A 2, {1} c, {3} c, {7} c, {9} c, X} = S A S B. Hence, the family S B S A is a semigroup of sets. Remark 3.2. The binary operation is not commutative, neither is associative, in general. This can be observed from the following example. Example 3.4 Let A and B be two non-empty subsets of X such that A B. Consider the families of sets A = {, A}, B = {B} and C = {, X}. Then we have A B = {B} while B A = {B, B \ A}. It is clear also that (A B) C = {, B, X} while A (B C) = {, A, B, X}. 3.2 Extension of a semigroup of sets via an ideal of sets In this section, we present a way of obtaining a new semigroup of sets from an old one by the use of an ideal of sets. Proposition 3.2. Let S be a semigroup of sets and I be an ideal of sets. Then the family S I is a semigroup of sets. Proof. Take two elements U 1 and U 2 of S I. Then U 1 = (S 1 \ I 1 ) I 1 and U 2 = (S 2 \ I 2 ) I 2 ) for some sets S i S and I i, I i I where i = 1, 2. We need to show that the set U = U 1 U 2 S I. In fact U = ((S 1 \ I 1 ) I 1 ) ((S 2 \ I 2 ) I 2 ) = (S 1 \ I 1 ) (S 2 \ I 2 ) (I 1 I 2 ). Since I is an ideal of sets, then the set I 2 = I 1 I 2 I. It follows that U = ((S 1 I c 1 ) (S 2 I c 2 ))cc I 2 =

3.2 Extension of a semigroup of sets via an ideal of sets 19 ((S 1 I c 1 )c (S 2 I c 2 )c ) c I 2 = ((S1 c I 1 ) (Sc 2 I 2 ))c I 2 = ((S1 c Sc 2 ) (Sc 1 I 2 ) (Sc 2 I 1 ) (I 1 I 2 ))c I 2. Put I 1 = (S1 c I 2 ) (Sc 2 I 1 ) (I 1 I 2 ) and note that U = ((S1 c Sc 2 )c I1 c) I 2 = ((S 1 S 2 ) I1 c) I 2 = ((S 1 S 2 ) \ I 1 ) I 2. It is easy to see that S 1 S 2 S and I 1, I 2 I. Hence, U S I. The following proposition shows the relationship in the sense of inclusion between the defined operations when they are applied on semigroups of sets and ideals of sets. Proposition 3.3. Let S be a semigroup of sets and I be an ideal of sets. Then (a) S I = S I S I = I S S; (b) (S I) I = S I, I (I S) = I S. Proof. (a) Note that for any set S S and for any set I I we have S I = (S \ I) (I \ S) S I, S I = S (I \ S) S I, S I = (I \ S) S I S and S = S S I. Thus, S I S I S I S and I S S I. Observe also that for any sets S 1, S 2 S and any sets I 1, I 2 I we have (S 1 \ I 1 ) I 2 = S 1 I S I, where I = ((I 1 S 1 ) \ I 2 ) (I 2 \ S 1 ), and (I 1 \ S 1 ) S 2 S I. There by, S I S I and I S S I. (b) Let S S and I 1, I 2, I 3, I 4 I. Observe that (((S \ I 1 ) I 2 ) \ I 3 ) I 4 = (S \ (I 1 I 3 )) ((I 2 \ I 3 ) I 4 ) S I. Hence (S I) I S I. The opposite inclusion is evident. Let I 1, I 2, I 3 I and S 1, S 2, S 3, S 4 S. Note that (I 1 \ ((I 2 \ S 1 ) S 2 )) ((I 3 \ S 3 ) S 4 ) = ((I 1 \ ((I 2 \ S 1 ) S 2 )) (I 3 \ S 3 )) S 4 = I S 4 I S, where I = (I 1 \ ((I 2 \ S 1 )) S 2 )) (I 3 \ S 3 ). Hence I (I S) I S. The opposite inclusion is evident. Corollary 3.1. Let S be a semigroup of sets and I be an ideal of sets. Then (a) The families S I, I S are semigroups of sets; (b) (I S) I = I (S I) = S I Proof. As S I = S I, by Proposition 3.2, the family S I is a semigroup of sets. From the observation (i) made above and the equality S I = I S, the family I S is also a semigroup of sets. This proves item (a). The item (b) follows by observing that S I = (S I) I (I S) I S I and S I = (S I) I I (S I) S I.

20 3 Algebra of semigroups of sets The following statement is evident. Corollary 3.2. Let I 1, I 2 be ideals of sets. Then the family I 1 I 2 is an ideal of sets. Moreover, I 1 I 2 = I 2 I 1 = I 1 I 2 = I 1 I 2. Example 3.5 Let X = {1, 2}, A = X, B = {1}, C = {2}, A = {A}, B = {B}. Note that S A = {A}, S B = {B}, I B = {, B}, S A I B = {A, C} and I B S A = {A}. Thus, in general, none of the following statements is valid: S I = I S, S I I, the family S I is an ideal of sets or I S is an ideal of sets, where S is a semigroup of sets and I is an ideal of sets. For two subfamilies A and B of P(X), put A B = {Y : Y A and Y B}. The next statement is useful in the search of pairs of semigroups of sets without common elements. Proposition 3.4. Let I be an ideal of sets and A, B P(X) such that: (a) A I = (i.e. A and I have no common element); (b) For each element U S A and each non-empty element B B there is an element A A such that A B \ U. Then (1) For each element I I, each element U S A and each non empty element B B we have (U I) c B = ; (2) For each elements I 1, I 2 I, each element U S A and each non-empty B B we have (U I 1 ) c (B \ I 2 ) = ; (3) For each elements I 1, I 2, I 3, I 4 I, each element U S A and each element V S B we have (U \ I 1 ) I 2 = (V \ I 3 ) I 4, i.e. (S A I) (S B I) =. Proof. (1) Assume that U I B for some non-empty element B B. By (b) there is A A such that A B \ U. Note that A (U I) \ U I. But this is a contradiction with (a). (2) Assume that U I 1 B \ I 2 for some non-empty element B B and some element I 2 I. Note that U (I 1 I 2 ) = (U I 1 ) I 2 (B \ I 2 ) I 2 B. But this is a contradiction with (1).

3.2 Extension of a semigroup of sets via an ideal of sets 21 (3) Assume that (U \ I 1 ) I 2 = (V \ I 3 ) I 4 for some elements U S A, V S B and I 3, I 4 I. If V =, then (U \ I 1 ) I 2 = I 4 and so U I 1 I 4. But this is a contradiction with with (a). Hence V =. Note that there is a non-empty element B B such that B V. Further observe that U I 2 (U \ I 1 ) I 2 = (V \ I 3 ) I 4 B \ I 3. But this is a contradiction with (2).

4 Semigroups of sets defined by Vitali selectors on the real line In this chapter, we present diverse semigroups of sets on the real line whose elements do not possess the Baire property. These semigroups will be constructed by the use of a concept of Vitali selectors rigorously defined in the coming section. The Vitali selectors are closely related to the classical Vitali sets on the real line (one should to replace the rationals Q by any countable dense subgroup Q of the reals R in the Vitali construction). They were considered in [3] and were called Γ-selectors of R there. It is known (cf. [3]) that each Vitali set (even each Vitali selector) of R does not possess the Baire property. This result was extended by V.A. Chatyrko [5] who showed that finite unions of Vitali sets on the real line also do not possess the Baire property. Furthermore, he observed that the family S V(Q) of all finite unions of Vitali sets, as well as the family S V(Q) M, where M is the family of all meager sets on the real line, are semigroups of sets, invariant under translations of R, and the elements of S V(Q) M also do not possess the Baire property. So in this chapter, we generalize Chatyrko s results via replacing the Vitali sets by Vitali Q-selectors, where Q is a countable dense subgroup Q of R, and the family M by any ideal of sets on R in the formulas S V(Q) and S V(Q) M. We study the relationship between the semigroups for different Q in the sense of inclusion. We observe that in the family of all semigroups of sets S V(Q), where Q is varied, there is no element which contains all others. Furthermore we 23

24 4 Semigroups of sets defined by Vitali selectors on the real line consider the family V sup of all Vitali selectors of R and the family S V sup which we call the supersemigroup of sets based on Vitali selectors. We will show that the supersemigroup S V sup also consists of sets without the Baire property and is invariant under translations of R. Let us note that the semigroup S V sup contains all semigroups S V(Q). The results of this chapter were mostly taken from the articles [4] and [6]. 4.1 Vitali selectors of the real line Let R be the real line and Q be a countable, dense in the real line subgroup of (R, +). For an element x R, denote by T x the translation of R by x, i.e. T x (y) = y + x for each element y R. If A is a subset of R and x R, we denote T x (A) by A x. Define the equivalence relation E on R as follows: for x, y R, let xey if and only if x y Q and let E α (Q), α I be the equivalence classes. Observe that I = c, where c is the cardinality of the continuum, and for each index α I and each x E α (Q) we have E α (Q) = Q x. So, each equivalence class E α (Q) is dense in R. Definition 4.1. A Vitali Q selector (shortly, a Vitali selector) of R is any subset V of R such that V E α (Q) = 1 for each α I. Note that a Vitali Q selector is a Vitali set [2], if Q is the set Q of rational numbers. We continue with simple facts about Vitali selectors. Proposition 4.1. Let V be a Vitali Q selector of R. (i) If q 1, q 2 Q and q 1 = q 2 then V q1 V q2 =. (ii) R = q Q V q. (iii) The set V is not meager in R. Proof. (i) Let x V q1 V q2 with q 1, q 2 Q and q 1 = q 2. Then x can be represented in two ways: x = y + q 1 = z + q 2 for some y, z V. But y z = q 2 q 1 Q implies that y and z are in the same equivalence class. Since V E α (Q) = 1 for all α I, then y = z. This implies that q 1 = q 2. We have a contradiction.

4.1 Vitali selectors of the real line 25 (ii) If x R, then x belongs to a unique equivalence class E α (Q). Let v α be the representatative of E α (Q) in V, i.e. V E α (Q) = {v α }. So, x v α = q for some q Q. It follows that x = v α + q V q. (iii) If V is a meager in R, then each V q, q Q, is a meager subset of R, as a translation is a homeomorphism. This implies that the real line R is covered by countably many meager sets, and hence it is meager. This is in a contradiction with the Baire Category Theorem. Lemma 4.1. For each Vitali Q selector V of R and each element x R, the set V x is also a Vitali Q selector of R. Proof. Let V be an arbitrary Vitali Q selector and let x R. Since for any different elements v 1, v 2 of V, we have (v 1 + x) (v 2 + x) = v 1 v 2 R \ Q, the elements v 1 + x and v 2 + x belong to different equivalence classes E α, α I. Consider a fixed α I and let v α be the element in V satisfying {v α } = E α (Q) V. So v α x E β (Q) for some β I. Note that there is one element v β in V such that {v β } = V E β (Q). Since v α x and v β belong to the same equivalence class E β (Q), there is a q Q such that v α x v β = q. So v β + x = v α q E α (Q). As v β + x V x, we have V x E α (Q) = 1 for all α I. The family of all Vitali Q selectors of R associated to the subgroup Q will be denoted by V(Q) and S V(Q) will denote the semigroup of sets generated by V(Q) (Chapter 3, Section 3.1). Proposition 4.2. The families V(Q) and S V(Q) are invariant under translations of R. Proof. Let V V(Q). By Lemma 4.1, it follows that for each element x R, we have V x V(Q). So, the family V(Q) is invariant under translations of R. Since the family S V(Q) consists of all finite unions of elements of V(Q), then it is also invariant under translations of R. Lemma 4.2. For each U S V(Q) and each non-empty open set O of R, there is an element V V(Q) such that V O \ U. Proof. Let U S V(Q) and O be a non-empty open set of R. So U = n i=1 V i where V i V(Q). To continue with the proof, first we show the following useful claim.

26 4 Semigroups of sets defined by Vitali selectors on the real line Claim 4.1.1. For each element α I, we have E α (Q) (O \ U) =. Proof. From the density of each equivalence class E α (Q) in the real line, we have E α (Q) O = ℵ 0 for all α I. By the definition of a Vitali Q selector, we have E α (Q) U = E α (Q) ( n i=1 V i ) = n i=1 (E α (Q) V i ) n for all α I. These two facts show that E α (Q) (O \ U) =. For each equivalence class E α (Q), α I, choose one element y α in the set E α (Q) (O \ U). The set V of such elements y α is a Vitali Q selector of R. Moreover, V O and V U =. Hence V O \ U. Let O be the family of all open subsets of R. Note that O is a semigroup of sets and S O = O. Proposition 4.3. Let I be an ideal of subsets of R. Then the following statements hold. (i) The families S V(Q), I S V(Q) and S V(Q) I are semigroups of sets such that S V(Q) I S V(Q) S V(Q) I. (ii) If V(Q) I =, then (S V(Q) I) (O I) =. In particular, S V(Q) (O I) =. (iii) If I is invariant under translations of R, then the families I S V(Q) and S V(Q) I are also invariant under translations of R. Proof. (i) The family S V(Q) is a semigroup of sets by Proposition 3.1. By Proposition 3.2 and Corollary 3.1, the families S V(Q) I and I S V(Q) are also semigroups of sets. The inclusion follows from Proposition 3.3. (ii) To prove the equality (S V(Q) I) (O I) =, we apply Proposition 3.4, together with Lemma 4.2. Namely, the families A and B in Proposition 3.4 are considered as the families V(Q) and O, respectively, and Lemma 4.2 plays the same role as condition (b) in Proposition 3.4. The particular case follows from the inclusion S V(Q) S V(Q) I. (iii) The invariance of the families I S V(Q) and S V(Q) I under translations of R follows from Proposition 4.2 and the assumption made on I.

4.1 Vitali selectors of the real line 27 Let M be the σ ideal of meager sets in R and let B p be the family of all subsets of R having the Baire property. Observe that V(Q) M = (Proposition 4.1 (iii)), B p = O M (see Definition 2.10) and M is invariant under translations of R. Corollary 4.1. The families S V(Q), M S V(Q) and S V(Q) M are semigroups of sets such that S V(Q) M S V(Q) S V(Q) M. They are invariant under translations of R, and consist of sets without the Baire property. In particular, the family S V(Q) consists of sets without the Baire property. Proof. Note only that by Proposition 4.3 (ii), we have the equality (S V(Q) M) (O M) =. Since B p = O M, then S V(Q) M P(R) \ B p. In particular, from the inclusion S V(Q) S V(Q) M, it follows that S V(Q) P(R) \ B p. Let I f (resp. I c, I cd or I n ) be the ideal of finite (resp. countable, closed and discrete or nowhere dense) subsets of R. Recall that for these ideals of sets, we have the inclusions I f I cd I c M and I cd I n M. Note that these ideals are invariant under translations of R. Corollary 4.2. Let I be I f, I c, I cd or I n. Then the families I S V(Q) and S V(Q) I are semigroups of sets such that S V(Q) I S V(Q) S V(Q) I. They are invariant under translations of R and consist of sets without the Baire property. Proof. The results follows from Proposition 4.3 and the mentioned above inclusions. Remark 4.1. From the inclusions I f follows also that: I cd I c M and I cd I n M, it (a) S V(Q) I f S V(Q) I cd S V(Q) I c S V(Q) M and S V(Q) I cd S V(Q) I n S V(Q) M. (b) I f S V(Q) I cd S V(Q) I c S V(Q) M S V(Q) and I cd S V(Q) I n S V(Q) M S V(Q). It is interesting to know which of the semigroups are equal.

28 4 Semigroups of sets defined by Vitali selectors on the real line Example 4.1 Let Q = Q. Observe that for each element A S V(Q) I f, we have A Q <. In fact, if A S V(Q) I f then A = (U \ I 1 ) I 2 where U S V(Q) and I 1, I 2 I f. Recall that U = n i=1 V i, where V i V(Q). Thus, A Q = ((U \ I 1 ) I 2 ) Q (U I 2 ) Q = (U Q) (I 2 Q) U Q + I 2 Q n + I 2 <. This implies that the semigroups of sets S V(Q) I f and S V(Q) I cd are not equal. In fact, let V V(Q) and let Z be the set of all integers. Since Z is a closed and discrete subset of R, we have V Z S V(Q) I cd. But (V Z) Q = ℵ 0. So, by the above observation, V Z / S V(Q) I f. Proposition 4.4. Let I be an ideal of sets such that V(Q) I =. Then each element of the family S V(Q) I is zero-dimensional. In particular, each element of the family S V(Q) is zero-dimensional. Proof. Let A S V(Q) I. Then A = (U \ F) E for some U S V(Q) and E, F I. The equality V(Q) I = implies that A =. Since = A R then 0 dim A 1, by the monotonicity property of dimension. Assume that dim A = 1. By the Brouwer-Dimension Theorem, there must exist a non-empty open set O in R such that O A. But A = (U \ F) E U E. By Lemma 4.2, there exists V V(Q) such that V O \ U. So, V O \ U (U E) \ U E which implies that V I. This is a contradiction. So dim A = 0. The particular case follows from the inclusion S V(Q) S V(Q) I. Corollary 4.3. Each element of the family S V(Q) M is zero-dimensional. Proof. The statement follows from Proposition 4.4. 4.2 Countable dense subgroups of R and generated semigroups Let Q 1 and Q 2 be subgroups of (R, +). Define Q 1 + Q 2 = {q 1 + q 2 : q i Q i, i = 1, 2}. We observe that the sum Q 1 + Q 2 is a subgroup of (R, +) and both Q 1 and Q 2 are subgroups of Q 1 + Q 2. Moreover,

4.2 Countable dense subgroups of R and generated semigroups 29 (i) If each Q i, i = 1, 2 is countable then the subgroup Q 1 + Q 2 is countable; (ii) If one of the subgroups Q i, i = 1, 2 is dense then so is Q 1 + Q 2. Let F be the family of all countable, dense in the real line subgroups of the additive group (R, +). Proposition 4.5. For each Q 1 F, there is a Q 2 F such that Q 1 Q 2. Proof. Let Q 1 F. Since Q 1 is a countable subset of R and the set R is uncountable, we have R \ Q 1 =. Consider an element x R \ Q 1 and set Q 2 = Q 1 + xz = {q + nx : q Q 1, n Z}, where Z is the additive group of all integers. It is clear that Q 2 is a countable subgroup of (R, +) and Q 1 Q 2. The subgroup Q 2 is dense on R (it contains a dense subset Q 1 of R) and hence Q 2 F. Moreover, Q 1 Q 2, since x Q 2 \ Q 1. Proposition 4.6. Let Q 1, Q 2 be elements of F such that Q 1 Q 2. Then S V(Q1 ) V(Q 2 ) =. In particular, V(Q 1 ) V(Q 2 ) =. Proof. Assume that there exists V S V(Q1 ) V(Q 2 ). Let E α (Q 2 ) be an equivalence class with the respect to the subgroup Q 2. So, V E α (Q 2 ) = 1 (4.1) Since Q 1 Q 2, we have Q 2 /Q 1 > 1, where Q 2 /Q 1 is the factor group of Q 2 by Q 1. Note that E α (Q 2 ) = β A α E β (Q 1 ), where E β (Q 1 ) are distinct equivalence classes with the respect to the subgroup Q 1 and A α = Q 2 /Q 1 > 1. It follows that V E α (Q 2 ) = V ( β A α E β (Q 1 )) = β A α (V E β (Q 1 )) = β Aα V E β = A α > 1. This is in contradiction with the Equality 4.1. So, we must have S V(Q1 ) V(Q 2 ) =. The particular case follows the inclusion V(Q 1 ) S V(Q1 ). Remark 4.2. Let Q 1, Q 2 be elements of F such that Q 1 Q 2 and Q 2 /Q 1 be the factor group of Q 2 by Q 1. We have either 1 < Q 2 /Q 1 < or Q 2 /Q 1 = ℵ 0. Proposition 4.7. For each Q 1 F there is a Q 2 F such that Q 1 Q 2 and Q 2 /Q 1 = ℵ 0. Proof. Let Q 1 F. We can apply repeatedly the Proposition 4.5 to obtain a sequence of elements in F in the following way:

30 4 Semigroups of sets defined by Vitali selectors on the real line Set Q 1 = Q 1. By Proposition 4.5 we can get Q 2 F such Q 1 Q 2, where Q 2 = Q 1 + x 1 Z, x 1 R \ Q 1. In the same way we can get Q 3 F such Q 2 Q 3 where Q 3 = Q 2 + x 2 Z, x 2 R \ Q 2, and so on. By this procedure, we get a sequence {Q k } k=1 of elements in F with Q 1 Q 2 Q 3 Q k 1 Q k, where Q k at the kth step is given by Q k = Q k 1 + x k 1 Z and x k 1 R \ Q k 1. The inequality Q k = Q k+1 is clear since x k Q k+1 \ Q k. Put Q 2 = k=1 Q k. It is evident that Q 2 is a subgroup of (R, +). Besides that, Q 2 is countable (it is a countable union of countable sets) and dense on R (it contains dense subsets Q k, k = 1, 2, of R). Hence Q 2 F. To prove that Q 2 /Q 1 = ℵ 0, we will observe that for each pair of distinct elements x n and x m in the sequence {x k } k=1, we have (Q 1 + x n ) (Q 1 + x m ) =. For, let y (Q 1 + x n ) (Q 1 + x m ) for some n > m. Then y = q 1 + x n = q 2 + x m for some q 1, q 2 Q 1. So x n = (q 2 q 1 ) + x m Q 1 + x m Q 1 + x m Z Q m+1. As n > m, we must have x n Q m+1 Q n. But, by the construction x n R \ Q n. We have a contradiction. So (Q 1 + x n ) (Q 1 + x m ) =. This observation implies that Q 1 + x 1, Q 1 + x 2,..., Q 1 + x k,... are different elements of Q 2 /Q 1. Hence, Q 2 /Q 1 = ℵ 0. Proposition 4.8. Let Q 1, Q 2 be elements of F such that Q 1 Q 2 and Q 2 /Q 1 = ℵ 0. Then S V(Q1 ) S V(Q2 ) =. Proof. Assume that there exists U S V(Q1 ) S V(Q2 ). Since U S V(Q 2 ), then U can be represented as U = n i=1 V i where V i V(Q 2 ) for all i. Let E α (Q 2 ) be an equivalence class with the respect to the subgroup Q 2. So, U E α (Q 2 ) n. (4.2) Since Q 2 /Q 1 = ℵ 0, we have E α (Q 2 ) = β A α E β (Q 1 ), where E β (Q 1 ) are distinct equivalence classes with the respect to the subgroup Q 1 and A α = ℵ 0. Since U S V(Q1 ) then U can be also represented as U = m k=1 W k where W k V(Q 1 ) for all k. Let W be an arbitrary Vitali Q 1 -selector among W i s making the union U. So U E α (Q 2 ) W E α (Q 2 ) = W ( β A α E β (Q 1 )) = β A α (W E β (Q 1 )) = β Aα W E β (Q 1 ) = A α = ℵ 0. This is in contradiction with the Inequality 4.2. So S V(Q1 ) S V(Q2 ) =.