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 of free Boolean algebras, and kappa-cc Boolean algebras. A central case lambda = (beth_omega)^+ or more generally, lambda = mu^+ for mu strong limit singular of ``small'' cofinality. A second one is mu = mu^{< kappa}< lambda < 2^mu ; the main case is lambda regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) etc.
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.
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…
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 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…
This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra $\algB$ and a property $Φ$ of measures one can recognize by purely combinatorial means if $\algB$ supports a strictly positive measure with property $Φ$. The most famous instance of this problem is MRP(countable additivity), and in the first part of the ar…
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…
The following results are proved: (a) In a model obtained by adding aleph_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 algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.
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 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-in…
We prove that there is a lattice embedded from every countable distributive lattice into the Boolean algebra of computable subsets of $\mathbb{N}$. Along the way, we discuss all relevant results about lattices, Boolean algebras and embeddings between them.
Constructing a fuzzy subset logic L with Boolean properties is notoriously difficult because under a handful of "reasonable" conditions, we have the following three debilitating constraints: (1) Bellman and Giertz in 1973 showed that if L is distributive, then it must be idempotent. (2) Dubois and Padre in 1980 showed that if L has the excluded middle or the non-contradiction property or both, then it must be non-idempotent. (3) Bellman and Giertz also demonstrated in 1973 that even if L is idempotent, then the only choice available for the (meet,join) logic operator pair is the (min,max) oper…
We study when a map between two subsets of a Boolean domain W can be extended to an automorphism of W. Under many hypotheses, if the underlying Boolean algebra is complete or if the sets are finite or Boolean domains, the necessary and suficient condition is that it preserves the Boolean distance between every couple of points.
The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…
In this note we derive a property of maximal ideal-independent subsets of boolean algebras which has corollaries regarding the continuum cardinals p and s_mm(P(omega)/fin).
In [Sh:89] we, answering a question of Monk, have explicated the notion of ``a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it'' (see section 2 here) and proved the existence of one with character density aleph_0, assuming first diamondsuit_{aleph_1} and then only CH. The idea was that if h is an endomorphism of B, not among the ``trivial'' ones, then there are pairwise disjoint D_n in B with h(d_n) not subset d_n. Then we can, for some S subset omega, add an element x such that d <= x for n in S, x cap d_n=0 for n not in S while forbidding a solution for {y…
We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(ω_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(ω)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.
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…
Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances i…
We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$.
The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of orde…
