Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unsupported Boolean algebras and forcing

If κ is an infinite cardinal, a complete Boolean algebra B is called κ‐supported if for each sequence 〈b_β : β < κ〉 of elements of B the equality _{α<κ β>α} b_β = equation/tex2gif-inf-5.gif _β∈A b_β holds. Combinatorial and forcing equivalents of this property are given and compared with the other forcing related properties of Boolean algebras (distributivity, caliber, etc.). The set of regular cardinals κ for which B is not κ‐supported is investigated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

P ≠ NP for all infinite Boolean algebras

We prove that all infinite Boolean rings (algebras) have the property P ≠ NP according to the digital (binary) nondeterminism.     

Some Boolean algebras with finitely many distinguished ideals II

We describe the countably saturated models and prime models (up to isomorphism) of the theory Th_prin of Boolean algebras with a principal ideal, the theory Th_max of Boolean algebras with a maximal ideal, the theory Th_ac of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.     

Iterations of Boolean algebras with measure

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

Special subalgebras of Boolean algebras

We consider eight special kinds of subalgebras of Boolean algebras. In Section 1 we describe the relationships between these subalgebra notions. In succeeding sections we consider how the subalgebra notions behave with respect to the most common cardinal functions on Boolean algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On poset Boolean algebras of scattered posets with finite width

We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.Mathematics Subject Classification (2000):Primary 03G05, 06A06, 06A11; Secondary 08A05, 54G12     

Projective free algebras of continuous functions on compact abelian groups

It is proved that the Wiener algebra of functions on a connected compact abelian group whose Bohr-Fourier spectra are contained in a fixed subsemigroup of the (additive) dual group, is projective free. The semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite. The projective free property is proved also for the algebra of continuous functions with the same condition on their Bohr-Fourier spectra. As an application, the connected components of the set of factorable matrices are described. The proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration.     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

The category of complete Boolean algebras is not an intersection of reflective subcategories of the category of frames

We answer a question of J. Rosický and W. Tholen by showing that the class of complete Boolean algebras (which is a prereflective subcategory of the category of frames) is not an intersection of reflective subcategories of the category of frames. In order to deduce this result, we first prove the following observation (using some basic facts about congruence frames): the three-element chain belongs to every reflective subcategory of the category of frames which contains the class of complete Boolean algebras.Supported by the Categorical Topology Research Group at the University of Cape Town, under funding from the Foundation for Research Development and the University of Cape Town.     

On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.

     

Finiteness conditions and distributive laws for Boolean algebras

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra are completely additive, he showed that the completion of always exists and is unique up to isomorphisms over . Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in must hold in the completion of . In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved. Received May 3, 1998; accepted in final form October 7, 1998.     

Countably categorical and autostable Boolean algebras with distinguished ideals

We study countable Boolean algebras with finitely many distinguished ideals (countable I-algebras) whose elementary theory is countably categorical, and autostable I-algebras which form their subclass. We propose a new characterization for the former class that allows to answer a series of questions about the structure of countably categorical and autostable I-algebras.