WHEN AN ATOMIC AND COMPLETE ALGEBRA OF SETS IS A FIELD OF SETS WITH NOWHERE DENSE BOUNDARY

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1 WHEN AN ATOMIC AND COMPLETE ALGEBRA OF SETS IS A FIELD OF SETS WITH NOWHERE DENSE BOUNDARY ARTUR BARTOSZEWICZ AND PIOTR KOSZMIDER Abstract. We consider pairs A, H(A) where A is an algebra of sets from some class called the class of algebras of type κ, λ and where H(A) is the ideal of hereditary sets of A. We characterize which of the above pairs are topological, that is, which are fields of sets with nowhere dense boundary for some topology together with the ideal of nowhere dense sets for this topology. Making use of the Balcar-Franek theorem we construct an example of a pair algebra, ideal with complete quotient algebra and the hull property but not topological. This countrexample, given in ZFC, provides the complete solution of the problem posed in [BBC]. Such an algebra was constructed in [B1] under some aditional set theoretic assumption. 1. Introduction For an ideal I and an algebra A of subsets of a set Y such that I A, we say that: the pair A, I has the hull property if and only if for every U Y there is a V A such that U V and for every W A if U W then V \ W I; the pair A, I is complete if and only if the quotient algebra A/I is complete; the pair A, I is topological if and only if there exists a topology τ on Y such that I is the ideal of nowhere dense sets and A is the algebra of all sets with nowhere dense boundary in the topology τ. The case when one considers only I and asks if it is the ideal of nowhere dense sets with respect to some topology was investigated in [CJ], where it is proved that it is always possible to find such a topology. Recently, one more related property of pairs A, I as above has been considered in several papers ([BBC], [BBRW], [BR], [BET], [ET]). Following the idea of Burstin and Marczewski, for any nonempty family F of nonempty subsets of Y, one can define the collections of sets: S(F) = {A Y : ( P F)( Q F)(Q A P or Q P \ A)} 1991 Mathematics Subject Classification. 28A05, 06E05, 03E35. Key words and phrases. Inner MB-representation; Isomorphism of Boolean algebras; Algebra of sets with nowhere dense boundary; Balcar-Franek theorem. 1

2 2 ARTUR BARTOSZEWICZ AND PIOTR KOSZMIDER and S 0 (F) = {A Y : ( P F)( Q F)(Q P \ A)} Then S(F) is an algebra and S 0 (F) is an ideal of subsets of Y (see [P, BBRW]). Adopting the notation as above, we say that the pair A, I has an inner MB-representation if and only if there exists an F A such that A = S(F) and I = S 0 (F). an algebra A is MB-representable if and only if A, I has an inner MB-representation for some ideal I of A. Using the results of [Bd], an example given in [BBC], and elementary properties of topological pairs we obtain the following implications between the introduced propeties: hull top hull & complete [Bd] [Bd] inner MB inner MB & complete complete The hull property and the completeness are independent, i.e. neither of them implies the other ([Bd]). The inner MB-representability does not imply the hull property ([BBC]). The implications between top and the remaining properties are elementary. It is proved in [B1] that the existence of a pair A, I which is complete and satisfies the hull property but is not topological, is consistent with ZFC. This is a partial solution of a problem posed in [BBC]. More precisely, we have shown the: Theorem 1. The following two conditions are equivalent: (I) there exists a set Y and an algebra A P(Y ) isomorphic to P(ω), with A, H(A) complete, inner MB-representable but not topological; (II) there exists an ideal J P(ω) such that P(ω)/J is isomorphic to P(ω 1 ). Here H(A) denotes the ideal of hereditary sets of A, i.e., sets whose power set is included in A. To show the consistency of (II), we have used an almost disjoint family of sets in P(ω), called strong-q-sequence. Steprāns in [S] has proved that the existence of a strong-qsequence with cardinality ω 1 is consistent. In January 2007 professor Balcar informed the first author that, by corollary from the Balcar-Franek theorem on independent families of sets in complete Boolean algebras, we can observe that the condition (II) is equivalent to the equality 2 ω = 2 ω 1 (and hence it holds f.e by Martin Axiom). In fact, it is enough to

3 use in this case the older and weaker version of Balcar-Franek theorem: the Fichtenholz- Kantorowitch theorem. In the present paper, using the full strengh of the Balcar-Franek theorem we give the complete answer to the title question originally posed in [BBC], i.e., we show in ZFC that there exists an algebra of sets A, such that A, H(A) is complete, has the hull property but is not topological. 2. Useful facts For the convenience of the reader we recall some basic facts and notions (presented in almost the same form in [B1]). Definition 1. Let F 1, F 2 P(Y ). We say that F 1, F 2 are coinitial if F 1 is dense in F 2 and vice versa, i.e. any set of any one of the families F 1, F 2 has a subset belonging to the other one. Fact 1 ([BBRW]). If F 1, F 2 are coinitial then S(F 1 ), S 0 (F 1 ) = S(F 2 ), S 0 (F 2 ). Conversely, if S(F 1 ), S 0 (F 1 ) = S(F 2 ), S 0 (F 2 ) and F i S(F i ) for i = 1, 2 then F 1, F 2 are coinitial. Fact 2 ([BBRW]). A, I is topological with respect to a topology τ on Y if and only if A, I = S(τ 0 ), S 0 (τ 0 ) where τ 0 = τ \ { }. Fact 3 ([BBC]). A, I is inner MB-representable if and only if A, I = S(A \ I), S 0 (A \ I). Moreover, if A, I is inner MB-representable then A, H(A) has the same property. Now, let us recall some basic algebraic considerations from [B2] concerning the structure of an isomorphism between P(X) (for some set X) and a field A of subsets of Y. Let Φ : P(X) A P(Y ) be such an isomorphism. Denote by Z the union of all images of the atoms of A, i.e. Z = Φ({x}). For any A P(X), denote x X 3 Φ 1 (A) = Φ(A) Z and Φ 2 (A) = Φ(A) \ Z. Then Φ 1 is an isomorphism between P(X) and some field of subsets of Z, and Φ 2 is an epimorphism from P(X) to some field B of subsets of Y \ Z. (We say that Φ 2 (A) is the remainder of Φ(A) and we call B the remainder algebra). We can describe Φ 1 by the formula Φ 1 (A) = x A Φ({x}) for A P(X). Denote by J the kernel of the homomorphism Φ 2. Then J = {A P(X) : Φ(A) = x A Φ({x})}.

4 4 ARTUR BARTOSZEWICZ AND PIOTR KOSZMIDER Note that J contains all finite subsets of X. Moreover, J = P(X) if and only if Z = Y, or (what is equivalent) if Φ 2 is the zero-homomorphism. By standard algebraic considerations we have that the field B of the remainders Φ 2 (A) for A P(X) is isomorphic to the quotient algebra P(X)/J. Observe, as in [B2], that for the algebra A = Φ[P(X)], we have H(A) = {Φ(A) : A J, x A Φ(x) = 1}. The symbols Z, Φ, Φ 1, Φ 2, J, B will retain their meaning throughout the paper. Note that given sets z y and a monomorphism φ 1 : P(X) z such that the images of singletons cover z and a homomorphism φ 2 : P(X) (y \ z) such that the kernel of φ 2 contains all finite subsets of X one can define an isomorphism onto its image Φ : P(X) P(y) by putting Φ(A) = φ 1 (A) φ 2 (A) for A X. Note that in such a case we have Φ i = φ i for i = 1, 2, Y = y, Z = z. In these cases we will use only use the notation Φ i, Y, Z instead of φ i, y and z for i = 1, 2. That is, unlike before, Φ will be obtained from Φ 1 and Φ 2. Fact 4 ([B2]). Let A = Φ[P(X)] be a field of subsets of Y where Φ is an isomorphism. Then A is inner MB-representable (and consequently, A, H(A) is inner MBrepresentable) if and only if the following two conditions are satisfied simultaneously: ( ) the set of all x X such that Φ({x}) > 1, belongs to J ; ( ) the remainder algebra B is atomic and the atoms of B cover Y \ Z. Moreover, if ( ) and ( ) are satisfied then A, H(A) = S(F), S 0 (F) for F = F 1 F 2 where F 1 = {Φ({x}) : Φ({x}) > 1}, and At(B) is the set of atoms of algebra B. F 2 = {Φ 1 (A) Φ 2 (A) : Φ 2 (A) At(B)} By Facts 1, 2, 4 we have: Fact 5. If A P(Y ) is isomorphic to P(X) then A, H(A) is topological if and only if A satisfies ( ) and ( ) and there exists a topology τ on Y such that τ \ { } is coinitial with F. Fact 6 ([B2]). If the algebra A is isomorphic to P(X) and A, H(A) has the hull property then the quotient algebras A/H(A) and P(X)/J are complete. Moreover if A satisfies (*), then the completness of A/H(A) and P(X)/J are equivalent.

5 By Facts 4, 5, 6 in the class of atomic, complete Boolean algebras of sets (i.e. in the class of algebras isomorphic to some power set P(X)) we can find topological pairs A, H(A) only if A satisfies ( ), ( ) and the remainder algebra B is isomorphic to P(λ) for some cardinal number λ. So it seems reasonable to introduce the following: Definition 2. We say that an algebra of sets A is of the type κ, λ for some cardinal numbers κ and λ (A κ, λ ) if (a) there exists an isomorphism Φ between P(X) an A for some set X with cardinality κ. (b) the conditions ( ) and ( ) are satisfied. (c) there exists an isomorphism Ψ between the remainder algebra B and P(λ). Observe that we can always take X = κ and that κ can be strictly less than λ as we can see in the following proposition Results Proposition 1. There exists an algebra A κ, λ if and only if 2 κ 2 λ. Proof. By the condition (c) of the last definition the inequality 2 κ 2 λ is necessary. So, let us assume this inequality. Then, by the Balcar-Franek theorem there exists an epimorphism Φ 2 from P(κ) onto P(λ) such that all finite subsets of κ are in its kernel (see [K]). Put Z = κ {0} and Y = Z λ. Let Φ : P(κ) P(Y ) be given by the formula Φ(A) = (A {0}) Φ 2 (A) Then J is the kernel of the homomorphism Φ 2. Since Φ({x}) = {x} {0} for each x κ we obtain ). Morover Y Z = λ and so A = Φ[P(κ)] satisfies all desired conditions. Definition 3. Let A κ, λ and {B ξ : ξ < λ} be the family of all atoms of B. We say that a family X = {X ξ : ξ < λ} of subsets of κ, generates atoms of B if for every ξ < λ we have Φ 2 (X ξ ) = B ξ. We say that X has a disjoint refinement generating atoms of B if there exists a family T = {T ξ : ξ < λ} such that for every ξ < λ we have T ξ X ξ, Φ 2 (T ξ ) = B ξ and the sets in T are pairwise disjoint. Proposition 2. For A κ, λ the following conditions are equivalent: (i) the pair A, H(A) is topological. (ii) for every family X generating atoms of B there exists a disjoint refinement with the same property. (iii) for some family X generating atoms of B there exists a disjoint refinement with the same property.

6 6 ARTUR BARTOSZEWICZ AND PIOTR KOSZMIDER Proof. (i) (ii). Let X be a family of subsets of κ generating atoms of B. Observe that such a family is almost disjoint with respect to the ideal J = Ker Φ 2 (i.e for α β we have X α X β J ). Denote by A 0 the set of all points η κ for which Φ({η}) > 1. By ( ) A 0 J, so the sets {X ξ \ A 0 : ξ < λ} form a family of sets generating atoms of B. Thus without lost of generality, we can assume that all the sets {X ξ : ξ < λ} are disjoint with A 0. Hence for all α < β < λ we have Φ 1 (X α ) Φ 1 (X β ) H(A) By Facts 4, 5, any set of the form Φ 1 (X ξ ) B ξ containes some subset G ξ open in τ, such that {Φ 1 (G ξ ) : ξ < λ} is a family generating atoms of B. Thus X ξ \ Φ 1 (G ξ ) J and consequently Φ 1 (X ξ ) \ G ξ H(A). So, we have G α G β H(A) for α β but H(A) consists of nowhere dense sets and thus G α G β =. The family {Φ 1 (G ξ ) : ξ < λ} is the disjoint refinement of X generating atoms. (ii) (iii) Obvious. (iii) (i) Let T be a disjoint refinement of X generating atoms of B. Put τ 1 = {Φ 1 (T ξ \ A) B ξ : ξ < λ, A J } and τ 2 = F 2 = {Φ({η}) : Φ({η}) > 1} Let τ be the topology generated by τ 1 τ 2. Then τ 0 is coinitial with F (definition 1). Proposition 3. If A κ, λ) and λ > κ then A, H(A) is not topological. Proof. There is no family of λ disjoint subsets of κ. Proposition 4. For any λ κ there exists an algebra of sets A κ, λ such that the pair A, H(A) is topological. Proof. Let us divide κ into λ disjoint subsets {X ξ : ξ < λ} with cardinality X ξ = κ. Let {J ξ : ξ < λ} be a family of maximal ideals on P(κ) with the following properties: [κ] <ω J ξ for any ξ < λ. X ξ / J ξ for any ξ < λ. Then we have: X α J β for α β. Let J = ξ<λ J ξ. Observe that A J if and only if for any ξ we have A X ξ J or (what is equivalent) if for any ξ we have A X ξ J ξ. Hence the following hold: (1) the class of equivalence [X ξ ] modulo J is an atom in P(κ)/J. (2) the quotient algebra P(κ)/J is atomic.

7 7 (3) the quotient algebra P(κ)/J is complete. The condition (3) follows from the disjointness of the family {X ξ : ξ < λ}. Thus the quotient algebra P(κ)/J is isomorphic to P(λ). Y = Z λ. Let Φ : P(κ) P(Y ) be given by the formula Φ(A) = (A {0}) Ψ([A]) Put Z = κ {0} and where Ψ is an isomorphism between P(κ)/J and P(λ). Then Φ 2 (X ξ ) = {ξ} for ξ < λ and {X ξ : ξ < λ}, as a disjoint family, obviously has a disjoint refinement with the same property. By Proposition 2 this completes the proof. Proposition 5. If κ λ > ω then there exists an algebra A κ, λ such that the pair A, H(A) is not topological. Proof. Let M be a complete Boolean algebra satisfying the ccc condition with cardinality M = 2 κ. For example M may be a measure algebra. For existence such an algebra see [F]. Then by the Balcar-Franek theorem there exists an epimorphisms f : P(κ) M whose kernel contains all finite subsets of κ and there exists an epimorphism g : M P(λ). Let Φ 2 be a composition Φ 2 = g f. Then Φ 2 is an epimorphism from P(κ) onto P(λ) whose kernel contains all finite subsets of κ. Put Z = κ {0} and Y = Z λ. Let us define Φ(A) = (A {0}) Φ 2 (A). As in the proof of proposition 1, one notes that J is equal to the kernel of Φ 2 and both ) and ) are satisfied concluding that A is of type κ, λ. For any ξ < λ let us define X ξ = Φ 1 2 ({ξ}). One can see that the family χ = {X ξ : ξ < λ} of subsets of κ generates atoms of B. However χ does not have a disjoint refinement {T ξ < ξ < λ} because f[t ξ ] s would be disjoint and non-zero, which is impossible since M is ccc and λ > ω. Theorem 2. There exists an algebra of sets A such that the pair A, H(A) is complete, has the hull property but is not topological. Proof. Let A be the algebra of the previous proposition. Since A is of type κ, λ, it is MB-representable by fact 4. It is clear from the construction that A/J is isomorphic to P(λ) and hence complete. Now, to conclude the theorem, use the result of [Bd] saying that an algebra A is MB-representable and complete if and only if it is complete and has the hull property. The above theorem is the complete solution of the problem posed in [BBC] is it possible to reverse the implication top hull & complete? The answer is NO and the countrexample is given in ZFC.

8 8 ARTUR BARTOSZEWICZ AND PIOTR KOSZMIDER References [BBC] M. Balcerzak, A. Bartoszewicz, K. Ciesielski, Algebras with inner MB-representation, Real. Anal. Exchange, 29(1), ( ), [BBRW] M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wroński, Marczewski fields and ideals, Real Anal. Exchange 26(2) ( ), [BR] M. Balcerzak, J. Rzepecka, Marczewski sets in the Hashimoto topologics for measure and category, Acta Univ. Carolin. Math. Phis. 39 (1998), [Bd] S. Baldwin, The Marczewski hull property and complete Boolean algebras, Real Anal. Exchange 28 (2002/2003), [B1] A. Bartoszewicz, On some algebra of sets in Steprāns strong-q-sequence model, Topology Appl. 149 (2005), no. 1-3, [B2] A. Bartoszewicz, Marczewski-Burstin representation of Boolean algebras isomorphic to a power set, Bull. Polish Acad. Sci. Math., 53 (2005), [BET] J.B. Brown, H. Elalaoui-Talibi, Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions, Colloq. Math. 82 (1999), [CJ] K. Ciesielski, J. Jasiński, Topologies making a given ideal nowhere dense or meager, Topology Appl. 63 (1995), [ET] H. Elalaoui-Talibi, On Marczewski-Burstin like characterizations of certain σ-algebras and σ-ideals, Real Anal. Exchange 26(1) ( ), [F] D. Fremlin, Measure algebras, vol.3. In Monk and Bonnet, Handbook of Boolean Algebras, North-Holland, Amsterdam [K] S. Koppelbeng, General theory of Boolean algebras, vol.1. In Monk and Bonnet, Handbook of Boolean Algebras, North-Holland, Amsterdam [P] J. Pawlikowski, Parametrized Ellentuck theorem, Topology Appl. 37 (1990), [S] J. Steprāns, Strong-Q-sequences and variations on Martin s Axiom, Canad. J. Math. 37(4), (1985), Institute of Mathematics, Lódź Technical University, ul. Wólczańska 215, Lódź address: arturbar@p.lodz.pl address: piotr.koszmider@gmail.com