Maharam algebras and Cohen reals

We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.

     

How special are Cohen and random forcings,i.e. boolean algebras of the family of subsets of reals modulo meagre or null

We prove that any Souslin c.c.c. forcing notion which adds a nondominated real adds a Cohen real. We also prove that any Souslin c.c.c. forcing adds a real which is not on any old “narrow” tree.     

Von Neumann's Problem and Large Cardinals

It is a well-known problem of Von Neumann to discover whetherthe countable chain condition and weak distributivity of a completeBoolean algebra imply that it carries a strictly positive probabilitymeasure. It was shown recently by Balcar, Jech and Pazák,and by Velikovi, that it is consistent with ZFC, modulo theconsistency of a supercompact cardinal, that every ccc weaklydistributive complete Boolean algebra carries a contiuous strictlypositive submeasure – that is, it is a Maharam algebra.We use some ideas of Gitik and Shelah and implications fromthe inner model theory to show that some large cardinal assumptionsare necessary for this result. 2000 Mathematics Subject Classification03E55, 28A60 (primary), 03E75 (secondary).     

Complete CCC Boolean Algebras, the Order Sequential Topology, and a Problem of Von Neumann

Let B be a complete ccc Boolean algebra and let _s be the topologyon B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonemptyopen set in (B, _s) is topologically dense.
2. It is consistentthat every weakly distributive complete cccBoolean algebracarries a strictly positive Maharam submeasure.
3. The topologicalspace(B, _s) is sequentially compact if and onlyif the genericextension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but notindependent reals. 2000 Mathematics Subject Classification 28A60,06E10 (primary); 03E55, 54A20, 54A25 (secondary).     

On cardinal collapsing with reals

We show that every uncountable regular cardinal can become {ie11-1} in a suitable Cohen extension by a real (set of integers) without destroying larger cardinals. The proof is analyzed to obtain results about the powers of Boolean algebras.     

Mathias–Prikry and Laver type forcing; summable ideals,coideals, and +-selective filters

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.     

A consistency result on thin-very tall Boolean algebras

It was proved by Baumgartner and Shelah that Con (ZFC)→Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω₂”). In this paper we improve Baumgartner-Shelah’s theorem, showing that Con (ZFC)→Con (ZFC+“for every α<ω₃ there is a superatomic Boolean algebra of width ω and height α”). The preparation of this paper was supported by DGICYT Grant PB98-1231.     

Ultra LI-ideals in lattice implication algebras

We define an ultra LI-ideal of a lattice implication algebra and give equivalent conditions for an LI-ideal to be ultra. We show that every subset of a lattice implication algebra which has the finite additive property can be extended to an ultra LI-ideal.     

Iterating ordinal definability

The paper presents a uniform way of obtaining by forcing descending sequences of the iterations of HOD and describes their structure. Among other things the following results are proved:Theorem. Any model M of ZFC can be obtained by the transfinite iteration of HOD in some generic extension N of M, i.e. M=(HOD^On)^N.Theorem (ZFC). For any ordinal α there is a complete Boolean algebra B with a decreasing sequence of derivatives of length α.Moreover some other results about ordinal definability are reproved using the introduced method. This method is a refinement of the classical method of coding introduced by McAloon in [11].     

Sequences of Generous and Nonatomic Quasi-Measures on Boolean Algebras

The notion of generosity, which is introduced in this paper, is closely connected with that of nonatomicity. We discuss the problem of when a sequence of generous [resp., nonatomic] quasi-measures or measures on a Boolean algebra or -algebra is, in some sense, uniformly generous [resp., nonatomic]. An application to sequences of vector-valued -additive functions is also included.     

Birkhoff Centre of a Semi Group

The concept of the Birkhoff centre of a semi group with 0 and 1 was introduced by U.M. Swamy and G.S. Murti [3], analogous to that of a bounded Poset [1], and proved that it is a Boolean algebra. This concept was extended to a semi group with sufficiently many commuting idempotents in [2] by G.S. Murti and proved that it is a relatively complemented distributive lattice. In this paper, we extend the above concept for a general semi group S and prove that the Birkhoff centre of any semi group S is a relatively complemented distributive lattice.AMS Subject Classification (1991): 06A12, 20M10.     

Ideals without ccc and without property

We prove a strong version of a theorem of Balcerzak-Roslanowski-Shelah by showing, in ZFC, that there exists a simply definable Borel -ideal for which both the ccc and property (M) fail. The proof involves Polish group actions.

     

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing-related properties of a complete Boolean algebra B with the properties of the convergences λ_s (the algebraic convergence) and λ_ls on B generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ_ls is a topological convergence iff forcing by B does not produce new reals and that λ_ls is weakly topological if B satisfies condition (?) (implied by the t-cc). On the other hand, if λ_ls is a weakly topological convergence, then B is a 2^h-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence λ_ls on the collapsing algebra \(B = {\text{ro}}{(^{ < \omega }}{\omega _2})\) is weakly topological” is independent of ZFC.     

Hechler's theorem for the meager ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing.     

Ultra <Emphasis Type="Italic">LI</Emphasis>-ideals in lattice implication algebras and <Emphasis Type="Italic">MTL</Emphasis>-algebras

A mistake concerning the ultra LI-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an LI-ideal to be an ultra LI-ideal are given. Moreover, the notion of an LI-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) LI-ideal and an ILI-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper LI-ideal and Boolean LI-ideal, (2) prime proper LI-ideal and ILI-ideal, (3) proper obstinate LI-ideal, (4) ultra LI-ideal. This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605389) and K. C. Wong Magna Fund in Ningbo University.     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

Molecules of Post algebras

Following A. Abian, we define amolecule of a partially ordered set to be any nonzero elementm with the property that any two nonzero elements less than or equal tom have a nonzero lower bound. In a Boolean algebra an element is a molecule iff it is an atom; however, in an arbitrary distributive lattice the notion of a molecule is too general to permit explicit characterization. In this note we examine the molecules in a coproduct of two distributive lattices and use our results to show that molecules play the same role in Post algebras as atoms do in Boolean algebras.Presented by R. S. Pierce.I would like to thank the referee for helpful comments that resulted in simplifying a number of proofs in this paper.     

A consistent very small Boolean algebra with countable automorphism group

It is consistent withZFC+¬CH that there is a Boolean algebra of cardinality ω₁ whose automorphism group is countably infinite. This answers a question of McKenzie and Monk.     

Strong Cardinals and Sets of Reals in Lω1(ℝ)

We generalize results of [3] and [1] to hyperprojective sets of reals, viz. to more than finitely many strong cardinals being involved. We show, for example, that if every set of reals in L_ω (?) is weakly homogeneously Souslin, then there is an inner model with an inaccessible limit of strong cardinals.