This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly sigma-filtered Boolean algebras. We show that for every uncountable regular cardinal kappa there are 2^kappa pairwise non-isomorphic openly generated Boolean algebras of size kappa > aleph_1 provided there is an almost free non-free abelian group of size kappa. The openly generated Boolean algebras constructed here are almost free.
Moreover, for every infinite regular cardinal kappa we construct 2^kappa pairwise non-isomorphic Boolean algebras of size kappa that are tightly sigma-filtered and c.c.c.
These two results contrast nicely with Koppelberg's theorem hat for every uncountable regular cardinal kappa there are only 2^{< kappa} isomorphism types of projective Boolean algebras of size kappa .
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A >>Cohen algebra<< is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.
Extending recent investigations on the structure of Tukey types of ultrafilters on $\mathcal{P}(ω)$ to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone's theorem from Boolean algebra.
How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many such endomorphisms in some reasonable ways. We prove that in any cardinality lambda=lambda^ {aleph_0} there is a Boolean algebra with no other endomorphisms. For this we use the so called "black boxes", but in a self contained way. We comment on how necessary the restriction on the cardinal is.
We study various forms of amalgamation for Boolean algebras with operations. We will also have the occasion to weaken the Boolean structure dealing with MV and BL algebras with operators.
We consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated.It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a in D the relative algebra B restriction a:= {b in B:b <= a} is isomorphic to B. In particular, the free product of countably many copies of an atomic Boolean algebra is homogeneous. Moreover, a Boolean algebra B is homogeneous if it satisfies the following conditions: (i) B has a countably generated ultrafilter…
We give a simplified proof of a theorem of M. Rabus and S. Shelah claiming that for each cardinal mu there is a c.c.c Boolean algebra with topological density mu.
We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -i…
In this paper we classify some special reducts of the countable atomless Boolean algebra which we call functional reducts. We prove that there are exactly $13$ such structures up to first order interdefinability.
The present status of the problems in my book "Cardinal Invariants on Boolean algebras" (Birkhauser 1996) is described, with a description of solutions or partial solutions, and references.
Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use skew BAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed skew BAs in terms of nBAs of n-partitions.
We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible.
We give some general theorems on free algebras of varieties of Boolean algebras with operators; a hitherto new result is obtained for Pinter's substitution algebras. For n\geq 3, and m>1, there is a generating set of the free algebra freely generated by m elements, which is not a free set of generators.
In this note we give an axiomatization of Boolean algebras based on weakly dicomplemented lattices: an algebra $(L,\wedge,\vee,\tu)$ of type $(2,2,1)$ is a Boolean algebra iff $(L,\wedge,\vee)$ is a non empty lattice and $(x\wedge y)\vee(x\wedge y\tu)=(x\vee y)\wedge(x\vee y\tu)$ for all $x,y\in L$. This provides a unique equation to encode distributivity and complementation on lattices.
We show that the Depth^+ of an ultraproduct of Boolean Algebras, can not jump over the Depth^+ of every component by more than one cardinality. We can have, consequently, similar results for the Depth invariant.
We present a new approach to ternary Boolean algebras in which negation is derived from the ternary operation. The key aspect is the replacement of complete commutativity by other axioms that do not require the ternary operation to be symmetric.
For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G. In particular, we will give a positive answer to the following question due to J.Roitman: Is aleph_{omega} a possible number of automorphisms of a rich Boolean algebra?
We introduce and examine order convergence and the interval topology on partially ordered sets in general. Problem 76 of Birkhoff's "Lattice Theory" asks whether for complete Boolean algebras the order topology and the interval topology coincide. We answer this question in the negative.
We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak A$ which is not in SPM but every subalgebra of $\mathfrak A$ of cardinality $\mathfrak c$ admits a strictly positive measure. This result is essentially due to Farah and Velickovic.
We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.
We give a necessary and sufficient condition for an atomless Boolean algebra to be countably generated, and use it to give new proofs of some some know facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor. We also show that Jensen's coding theorem can be used to provide cardinal preserving countably generated complete Boolean algebras of arbitrary large cardinality. This answers a question of Jech [5] from 1976.
The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphis…
The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds classically to that of map preserving arbitrary joins; we provide a description of atomic set-based overlap algebras in the language of formal topology, thus giving a predicative characterization of discrete locales; we show that the power-collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap alge…
If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{…
We present a collection of observations and results concerning submeasures on Boolean algebras. They are all motivated by Maharam's problem and Talagrand's construction that solved it.
We investigate the sequential topology $τ_s$ on a complete Boolean algebra $B$ determined by algebraically convergent sequences in $B$. We show the role of weak distributivity of $B$ in separation axioms for the sequential topology. The main result is that a necessary and sufficient condition for $B$ to carry a strictly positive Maharam submeasure is that $B$ is ccc and that the space $(B,τ_s)$ is Hausdorff. We also characterize sequential cardinals.
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean valued models and full details of all proofs are given. We focus our presentation using the boolean algebra language and defining an iteration system as a directed and commutative system of complete and injective homomorphisms between complete and atomless boolean algebras. It is well known that the boolean algebra approach to forcing and iterations is fully e…
A subset $A$ of a Boolean algebra $B$ is said to be $(n,m)$-reaped if there is a partition of unity $P \subset B$ of size $n$ such that the cardinality of $\{b \in P: b \wedge a \neq \emptyset\}$ is greater than or equal to $m$ for all $a\in A$. The reaping number $r_{n,m}(B)$ of a Boolean algebra $B$ is the minimum cardinality of a set $A \subset B\setminus \{0\}$ such which cannot be $(n,m)$-reaped. It is shown that, for each $n \in ω$, there is a Boolean algebra $B$ such that $r_{n+1,2}(B) \neq r_{n,2}(B)$. Also, $\{r_{n,m}(B) : \{n,m\}\subseteqω\}$ consists of at most two consecutive integ…
We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting.
The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i<delta >, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product prod_{i< delta} Dom(I_i), the existence proofs are related to pcf. The second theme is when does a Boolean algebra B has free caliber lambda (i.e. if X subseteq B and |X|= lambda, then for some Y subseteq X with |Y|= lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product o…
We prove the consistency of irr(prod limits_{i<kappa}B_i/D)< prod limits_{i<kappa}irr(B_i)/D, where D is an ultrafilter on kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk's list of problems, that is number 35. The solution applies to many other properties, e.g., Souslinity. Next, we get similar results with kappa = aleph_1 (easily we cannot have it for kappa=aleph_0) and Boolean Algebras B_i (i< kappa) of cardinality < beth_{omega_1}.
We give a unified treatment of the model theory of various enrichments of infinite atomic Boolean algebras, with special attention to quantifier-eliminations, complete axiomatizations and decidability. A classical example is the enrichment by a predicate for the ideal of finite sets, and a novel one involves predicates giving congruence conditions on the cardinality of finite sets. We focus on three examples, and classify them by expressive power.
We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the former case of independence can be seen as the discrete version of the latter.
We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP…
Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations
Czech Academy of Sciences（捷克科学院）；Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机研究所）；University of Silesia in Katowice（卡托维兹西里西亚大学）；Institute of Mathematics Cardinal Stefan Wyszyński University in Warsaw（华沙斯特凡·维辛斯基红衣主教大学数学研究所）
We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a ``black box'' at the end. The proof involves a vari…
Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardi…
Moisil in 1941, while constructing the algebraic models of n-valued Łukasiewicz logic defined the set $B^{[n]}$,where $B$ is a Boolean algebra and `n' being a natural number. Further it was proved by Moisil himself the representations of n-valued Łukasiewicz Moisil algebra in terms of $B^{[n]}$. In this article, structural representation results for Stone, dual Stone and double Stone algebras are proved similar to Moisil's work by showing that elements of these algebras can be looked upon as monotone ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and …
In this paper, we investigate related properties of some particular derivations and give some characterizations of additive derivations in MV-algebras. Then, we obtain that the fixed point set of additive derivations is still an MV-algebra. Also, we study boolean additive derivations and their adjoint derivations. In particular, we get that the fixed point set of boolean addition derivations and that of their adjoint derivations are isomorphism. Moreover, we prove that every MV-algebras are isomorphic to the direct product of the fixed point set of boolean additive derivations and that of thei…
We introduce quantum Boolean algebras which are the analogue of the Weyl algebras for Boolean affine spaces. We study quantum Boolean algebras from the logical and set theoretical viewpoints.
This paper illustrates the relationship between boolean propositional algebra and semirings, presenting some results of partial ordering on boolean propositional algebras, and the necessary conditions to represent a boolean propositional subalgebra as equivalent to a corresponding boolean propositional algebra. It is also shown that the images of a homomorphic function on a boolean propositional algebra have the relationship of boolean propositional algebra and its subalgebra. The necessary and sufficient conditions for that homomorphic function to be onto-order preserving, and also an extensi…
We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ^+,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ,μ)-FN for λ>μform a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The (κ,\aleph_0)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in t…
