Semigroups of pathological sets

Abstract In this article, we demonstrate how the Vitali and Bernstein constructions together with a simple theory on semigroups and ideals of sets can be used for producing different semigroups of sets without the Baire property and/or non-measurable in the Lebesgue sense.


Introduction
The family B p (ℝ) of subsets of the real line ℝ with the Baire property and the family L(ℝ) of subsets of ℝ measurable in the Lebesgue sense are well known in topology and analysis (cf. [5,7,8,12], etc). They have a number of similarities. Thus the families B p (ℝ) and L(ℝ) are σ-algebras of sets on ℝ, i.e., they are closed under all basic set theoretic operations. Moreover, B p (ℝ) is the minimal σ-algebra of sets which contains the family B(ℝ) of all Borel subsets of ℝ and all meager sets on ℝ, while L(ℝ) is the minimal σ-algebra of sets which contains the family B(ℝ) and all subsets of ℝ having the Lebesgue measure equal to zero (null sets). None of the families contains Vitali sets as well as Bernstein sets of the real line. Furthermore, the families B p (ℝ) and L(ℝ) are invariant under action of the group T(ℝ) of all translations of ℝ, i.e., if A ∈ B p (ℝ) (resp. A ∈ L(ℝ)) and h ∈ T(ℝ), then h(A) ∈ B p (ℝ) (resp. h(A) ∈ L(ℝ)).
Despite these similarities, the families B p (ℝ) and L(ℝ) are incomparable by inclusion. Indeed, recall (cf. [12]) that each subset of ℝ can be decomposed into a disjoint union of a first category set and a null set. So if we decompose a Vitali subset (or a Bernstein subset) of ℝ into a disjoint union of a meager set and a null set, then one of the sets of this decomposition must be Lebesgue non-measurable but meager, and the other must be a null set but without the Baire property. Thus none of the families B p (ℝ) and L(ℝ) contains the other (the same is evidently valid for their complements B c p (ℝ) and L c (ℝ) in the family P(ℝ) of all subsets of ℝ). Let us mention a difference between B p (ℝ) and L(ℝ) (resp. B c p (ℝ) and L c (ℝ)). The first family is evidently invariant under self-homeomorphisms of the real line, but the second one is not. Now we will name a similarity of the families B c p (ℝ) and L c (ℝ) which is fundamental for us. Unlike the pair B p (ℝ) and L(ℝ), the families B c p (ℝ) and L c (ℝ) are not closed under such basic set theoretic operations as the union and intersection of sets. So, in the study of B c p (ℝ) and L c (ℝ), various authors pay more attention to their elements with very curious properties (cf. [5]) than to the families themselves. Differently, we are interested in the properties of the family B c p (ℝ) (resp. L c (ℝ)). Thus, in [4] (see also [11]), the following project was started.
Look for sufficiently rich subfamilies of B c p (ℝ) (resp. L c (ℝ) or B c p (ℝ) ∩ L c (ℝ)) which have some algebraic structures and which are invariant under action of infinite subgroups of the group H(ℝ) of all homeomorphisms of ℝ. Example 1.1. Consider a subgroup G of the additive group (ℝ, +) of reals which is a Bernstein set (hence G is an element of B c p (ℝ) ∩ L c (ℝ)) and for which cardinality of the factor group (ℝ, +)/G is continuum c (cf. [5]). Then let F be the family of all cosets of G in ℝ. So F is invariant under translations of ℝ. Since elements of F are Bernstein sets (as translations of G), F is a subfamily of B c p (ℝ) ∩ L c (ℝ). Moreover, F is an abelian group with the group operation " * " defined as follows. For F 1 , F 2 ∈ F such that F 1 = x 1 + G and F 2 = x 2 + G for some It is natural to ask if we can find subfamilies of B c p (ℝ) (resp. L c (ℝ) or B c p (ℝ) ∩ L c (ℝ)) endowed with some elementary set theoretic operation(s).

Proposition 1.2.
Let H 1 be a non-trivial subgroup of H(ℝ), and let A be a subfamily of B c p (ℝ) (resp. L c (ℝ)) invariant under action of H 1 . Assume that, for each n ≥ 2 and each A 1 , . . . , . Then the family S A consisting of all unions of finitely many elements of A is an abelian semigroup of sets with respect to the operation union of sets. Moreover, S A is invariant under action of H 1 , and ). In addition, the dual family S * A = {ℝ \ A : A ∈ S A } is also an abelian semigroup of sets with respect to the operation intersection of sets. Moreover, Let us note that the Bernstein sets as well as the Vitali sets provide families satisfying Proposition 1.2 (see below). Example 1.3. The family F ⊆ B c p (ℝ) ∩ L c (ℝ) considered in Example 1.1 is invariant under translations of the real line, and for each n ≥ 2 and each F 1 , . . . , F n ∈ F we have ⋃ n i=1 F n ∈ B c p (ℝ) ∩ L c (ℝ) (the union is evidently a Bernstein set). So the family S F consisting of all unions of finitely many elements of F is an abelian semigroup of sets with respect to the operation union of sets. Moreover, S F is invariant under translations of ℝ and S F ⊆ B c p (ℝ) ∩ L c (ℝ). Similarly, the dual family S * F = {ℝ \ F : F ∈ S F } is also an abelian semigroup of sets with respect to the operation intersection of sets which is invariant under translations of ℝ and for which S * F ⊆ B c p (ℝ) ∩ L c (ℝ).  [3] and [6], respectively). So the family S V consisting of all unions of finitely many elements of V is an abelian semigroup of sets with respect to the operation union of sets. Moreover, S V is invariant under translations of ℝ and is also an abelian semigroup of sets with respect to the operation intersection of sets which is invariant under translations of ℝ and for which S * Let us point out some evident differences between the semigroups S F and S V from the examples above. As we already observed, each element of the family S F is a Bernstein set, but elements of S V are neither Bernstein sets nor Vitali sets as a rule. Furthermore, each element of S F as a Bernstein set must be everywhere dense in ℝ; however, each element of S V does not need to be everywhere dense in ℝ.
In this article, we will demonstrate how the ideas of Proposition 1.2, Examples 1.3, 1.4 and some simple theory on semigroups and ideals of sets can be used for producing various rich semigroups of sets without the Baire property (see also [1,4,11]) and/or non-measurable in the Lebesgue sense. The mutual relationship between the semigroups of sets are also under our consideration.

Elementary algebraic objects in set theory
By an algebraic property on a family of sets we mean any elementary set theoretic operation which is closed on the family. We continue with several known types of families of sets having algebraic properties.

Rings and algebras
Let X be a non-empty set and let P(X) be the family of all subsets of X. A non-empty family R ⊆ P(X) is called a ring of sets on X if A∆B ∈ R and A ∩ B ∈ R whenever A, B ∈ R.
Since A \ A = 0 for any A ∈ R, each ring of sets contains the empty set 0. Let A ⊆ P(X) be a ring of sets on X. The family A is called an algebra of sets on X if X ∈ A. In particular, for each element A ∈ A, its complement A c = X \ A in X is also an element of A. A ring R of sets on X is called a σ-ring of sets on X if ⋃ ∞ n=1 A n ∈ R whenever A i ∈ R for each i ≥ 1. A σ-ring of sets A on X is called a σ-algebra of sets on X if X ∈ A. It is easy to see that each σ-algebra of sets on X is also closed under countable intersections of sets. Let A ⊆ P(X). The smallest σ-ring (resp. σ-algebra) of sets on X containing A is called the σ-ring (resp. the σ-algebra) of sets on X generated by the family A.

Ideals
A non-empty family I ⊆ P(X) is called an ideal of sets on X if I satisfies the following conditions: (a) if A ∈ I and B ∈ I, then A ∪ B ∈ I; (b) if A ∈ I and B ⊆ A, then B ∈ I. Let us note that each ideal of sets on X contains the empty set 0 as an element. An ideal of sets I on X is called a σ-ideal of sets on X if it is closed under countable unions. Example 2.1. Consider some ideals of sets. (i) The family of all finite subsets of a set X is an ideal of sets on X. We will denote the ideal by I f . (ii) The family of all bounded subsets of the reals ℝ is an ideal of sets on ℝ. We will denote the ideal by I b .
(iii) The family of all countable subsets of a set X is a σ-ideal of sets on X. We will denote the ideal by I c .
(iv) Let A ⊆ X. Then the family I A = {B : B ⊆ A} is a σ-ideal of sets on X.
Let I be an ideal of sets on a set X and A, B ⊆ X. We say (cf. [8]) that A = B modulo I (in brief, A = B mod I) if A \ B and B \ A are elements of I (or, equivalently, A∆B ∈ I).
The following statement shows how to produce from one set outside an algebra a family of sets outside the algebra.
Let us recall that a non-empty set S is called a semigroup if there is a binary operation ⋆ : Since the operation union (resp. intersection) of sets is associative, we have the following natural definition. A non-empty family S ⊆ P(X) of sets is called a semigroup of sets with respect to the operation union (resp. intersection) of sets if, for each pair of elements A, B ∈ S, we have A ∪ B ∈ S (resp. A ∩ B ∈ S). Let us note that the semigroups of sets are abelian.
A semigroup of sets with respect to the operation union (resp. intersection) of sets on X is called a σ-semigroup of sets with respect to the operation union (resp. intersection) of sets if S is closed under unions (resp. intersections) of countably many sets.
Note that if a proper subfamily S of P(X) is a (σ-)semigroup of sets with respect to the operation union of sets, then the dual family S * = {X \ S : S ∈ S} is also a (σ-)semigroup of sets with respect to the operation intersection of sets, and vice versa.
In this article, we will discuss mostly (σ-)semigroups of sets with respect to the operation union of sets (shortly, (σ-)semigroups of sets).
, and note that the family S

(σ)
A is a (σ-) semigroup of sets on X. We will call S

(σ)
A a (σ-)semigroup of sets on X generated by the family A.
A a (σ-)ideal of sets on X generated by the family A.  Remark 2.7. If I 1 , I 2 are (σ-)ideals of sets, then the family I 1 * I 2 is an (σ-)ideal of sets. Moreover, we have I 1 * I 2 = I 2 * I 1 = I 1 ∆I 2 = I 1 ∪ I 2 .

New semigroups of sets from old by ideals of sets
Simple conditions allowing to distinguish the semigroups S∆I and S ∪ I can be found in the following statement.
The following statement can be used in the constructions of various semigroups of sets without the Baire property.

Proposition 2.9 ([1]). Let I be an ideal of sets, and let
3 Facts from measure theory and category

Lebesgue measurable sets on the real line
Recall (cf. [12]) that the outer measure m * of a subset A of the real line ℝ is defined by the formula Let us recall that the outer measure m * possesses the following evident properties: Let us note that, besides the mentioned earlier properties (a)-(d) of m * , for the function m, we have additionally the following: if A 1 , A 2 , . . . are disjoint measurable sets and

Meager sets in topological spaces
Let X be a topological space and A ⊆ X.
Note that the family of all closed and discrete subsets of X forms an ideal of sets. We will denote the ideal by I cd .
A subset A of X is said to be nowhere dense in X if Int X (Cl X (A)) = 0. Note that the family of nowhere dense sets in a given topological space is an ideal of sets. We will denote the ideal by I n .
A subset A of X is said to be dense in X if Cl X (A) = X. It is easy to see that if a set A is a nowhere dense subset of X, then the set X \ A is dense in X, and if a set B is open and dense in X, then X \ B is nowhere dense in X.
A subset A of X is said to be meager in X if A is the union of countably many nowhere dense subsets of X. Let us note that the family of all meager sets in the space X is a σ-ideal of sets on X. We will denote the ideal by M(X).

Remark 3.2.
For the real line ℝ, we have the following: Since the standard Cantor set ℂ on the interval [0, 1] is an uncountable nowhere dense subset of ℝ, while the set ℚ of rationals numbers is a countable dense subset of ℝ, the ideals I c and I n are not comparable in the sense of inclusion.

Sets with Baire property in topological spaces
Let X be a topological space with the topology τ X .
A subset A of X is said to have the Baire property in X if A = O∆M for some O ∈ τ X and some M ∈ M(X). Denote the family of all sets having the Baire property in a topological space X by B p (X). Proposition 2.2 easily implies the following.
Similarly, one can introduce and describe the sets with the Baire property on a topological space via closed sets of the space. Since each open (resp. closed) subset of X and each meager subset of X are elements of B p (X), it follows from Proposition 3.4 that B p (X) is the smallest σ-algebra of sets in X containing all open (resp. closed) and all meager subsets of X. In particular, we have B(X) ⊆ B p (X).

Vitali selectors
Below Q is any countable dense subgroup of the additive group (ℝ, +).
Following [5] (see also [11]), let us consider the following equivalence relation E(Q) on ℝ: for x, y ∈ ℝ, let xE(Q)y if and only if x − y ∈ Q. Denote by E α (Q), α ∈ I, the equivalence classes of the relation. Observe that |I| = c, where c is the cardinality of continuum, and for each α ∈ I and each x ∈ E α (Q), we have E α (Q) = Q + x (so E α (Q) is dense in ℝ).
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. If Q is the group ℚ of rational numbers, then the Vitali Q-selectors of ℝ are Vitali sets [13]. It is easy to see that, for each point p ∈ ℝ, there exists a Vitali Q-selector which contains the point p, and for each Vitali Q-selector V we have |V| = c.

Corollary 4.3. A closed subset A of ℝ contains a Vitali selector if and only if Int
Proof.    The following statement is important for Example 1.1.

Bernstein sets
Proposition 4.11 (cf. [5]). There exist two subgroups G 1 and G 2 of the additive group (ℝ, +) such that Remark 4.12. The family of all cosets of the subgroup G 1 (or G 2 ) from Proposition 4.11 forms a decomposition of the reals into continuum many Bernstein sets (as translations of G i ). Let us note that there exist decompositions of the real line ℝ into countably (resp. finitely) many Bernstein sets with different arithmetic properties (see [10]).
Let us note that there are Vitali sets which are no Bernstein sets, and Bernstein sets which are no Vitali sets.
Proposition 4.13 (cf. [5]). There is a subset A of ℝ such that A is both a Vitali set and a Bernstein set.

Remark 5.2.
Let us note that all elements of S σ V Q besides the set ℝ are without the Baire property [3].

Lemma 5.3. For each Vitali Q-selector V and each element x ∈ ℝ, the set V + x is also a Vitali Q-selector.
Let V(Q) be the family of all Vitali Q-selectors, and let S V(Q) (resp. S σ V(Q) ) be the semigroup (resp. the σsemigroup) of sets generated by V(Q).
It is easy to see that, for each Vitali Q-selector V, we have Moreover Proof. We will show the sufficiency of the first statement. Indeed, for each α ∈ I, let A ∩ E α (Q) = {x α 1 , x α 2 , . . . } (some elements in the sequence can be equal).
Corollary 5.6. The following statements hold.
In order to get the inclusion I ⊆ S V(Q) ∆I, apply Proposition 2.6. Note that, for any set U ∈ S V(Q) , we have |U ∩ E α (Q)| < ℵ 0 , but for U ∪ A ∈ S V(Q) ∪ I, we have |(U ∪ A) ∩ E α (Q)| = ℵ 0 .
(ii) The invariance of the families S V(Q) ∪ I and S V(Q) ∆I under translations of ℝ follows from Proposition 5.4 and the assumption made on I.  Proof. In order to get the equality (S V(Q) ∆I) ∩ (τ ℝ ∆I) = 0, apply Proposition 2.9 together with Lemma 5.9. Namely, consider the families V(Q) and τ ℝ as the families A and B of Proposition 2.9, respectively. Lemma 2.9 plays the same role as condition (ii) of Proposition 2.9.
Let us prove the second part of the statement. Let A ∈ S V(Q) ∆I. It follows from Proposition 2.6 that A = (U \ F) ∪ E for some U ∈ S V(Q) and E, F ∈ I. Since V(Q) ∩ I = 0, we have U \ F ̸ = 0. Hence A ̸ = 0. Moreover, 0 ≤ dim A ≤ 1.

Assume that dim
which implies that V ∈ I. This is a contradiction. Hence dim A = 0.
Lemma 5.21. The following inequalities hold: Proof. Consider any V ∈ V(Q).
(i) Indeed, let Z be an infinite closed discrete subset of ℝ consisting of elements of Q. Since Z ∈ I cd , we (ii) Let K be an infinite countable subset of Q with only one limit point 0 ∈ K. Since K ∈ I n , we have V ∪ K ∈ S V(Q) ∪ I n . Note that (V ∪ K) ∩ Q ⊇ K and K ∉ I cd . Hence we have (V ∪ K) ∩ Q ∉ I cd . It follows from Lemma 5.19 that V ∪ K ∉ S V(Q) ∆I cd .
(ii) Let {α i } ∞ i=1 be an infinite countable subset of I, the indexed set of the equivalence classes. For each positive integer i, consider a point x i from E α i (Q) ∩ (i, i + 1). The set S = {x 1 , x 2 , . . . } is required.
(iii) Choose as K the set (V + q 0 ) ∩ ℂ from Lemma 5.20, and put V K = V + q 0 .
Proof. Note that V S \ S ∈ S V(Q) ∆I cd ⊆ S σ V(Q) ∆I cd , where S and V S are from Lemma 5.23 (ii). Since, for each I ∈ I f , we have |I| < ℵ 0 , by Lemma 5.23 where K and V K are from Lemma 5.23 (iii). Since, for each I ∈ I c , we have |I| ≤ ℵ 0 , by Lemma 5.23 (i), we get V K \ K ∉ S σ V(Q) ∆I c .

Countable dense subgroups ofℝ and related semigroups of sets
Let F be the family of all countable dense, in the real line ℝ, subgroups of the additive group (ℝ, +).
Proof. Let Q 1 ∈ F. By Proposition 5.25, one can find Q 2 ∈ F such that Q 1 ⊊ Q 2 and |Q 2 /Q 1 | = ℵ 0 . It follows from Proposition 5.26 that S V(Q 2 ) ∩ S V(Q 1 ) = 0. So there is no Q * ∈ F such that the generated semigroup S V(Q * ) contains the semigroup S V(Q) for each Q ∈ F. (ii) For each Q ∈ F and each ideal of sets I on ℝ, the following inclusions hold:

Supersemigroup of sets based on all Vitali selectors
Similarly to Proposition 5.8, one can get the following. Similarly to Proposition 5.10, one can prove the following.
Proposition 5.31. Let I be an ideal of subsets on ℝ and V sup ∩ I = 0. Then (S V sup ∆I) ∩ (τ ℝ ∆I) = 0. In particular, S V sup ∩ (τ ℝ ∆I) = 0. Moreover, each element of the semigroup S V sup ∆I is zero-dimensional. In particular, each element of S V sup is zero-dimensional.
Since V sup ∩ M(ℝ) = 0, we have the following. Corollary 5.33 (cf. [11]). Let I be I f , I c , I cd or I n . Then the families S V sup ∪ I, S V sup ∆I are semigroups of sets, and S V sup ⊆ S V sup ∪ I ⊆ S V sup ∆I ⊆ S V sup ∆M(ℝ). The considered semigroups are invariant under translations of ℝ and consist of zero-dimensional sets without the Baire property.
Question 5.34. What is the relationship between the semigroups mentioned in Corollary 5.33?

Semigroups of Lebesgue non-measurable sets
In [6], Kharazishvili proved that every non-empty union of finitely many Vitali sets is not Lebesgue measurable, i.e., S V(ℚ) ⊆ L c (ℝ). Let us generalize the fact by the use of his method.