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.

An analogue of Mathias forcing is studied in connection of free Boolean algebras and nowhere dense ultrafilters. Some applications to rigid Boolean algebras are given.

We study L∞κ-freeness in the variety of Boolean algebras. It is shown that some of the theorems on L∞κ-free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L∞κ-free Boolean algebras which have no counterpart in the varieties above.

In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory . We show that any two countable atomless Boolean algebras are isomorphic and we give an example of such a Boolean algebra. One other main result is, that there is an infinite Boolean algebra freely generated by a set. At the end of the paper we show that the sentence “There is no non-trivial free group which is a set” is consistent with AST.

A generalization of conventional Horn clause logic programming is proposed in which the space of truth values is a pseudo-Boolean or Heyting algebra, whose members may be thought of as evidences for propositions. A minimal model and an operational semantics is presented, and their equivalence is proved, thus generalizing the classic work of Van Emden and Kowalski.

We study the problem of finding a basis for all rules admissible in the intuitionistic propositional logic IPC. The main result is Theorem 3.1 which gives a basis consisting of all rules in semi-reduced form satisfying certain specific additional requirements. Using developed technique we also find a basis for rules admissible in the logic of excluded middle law KC.

In this paper we will prove that ifF is a filter of a free Boolean algebra such that the minimal cardinality of the set of generators ofF is an uncountable regular cardinal or a singular cardinal with uncountable cofinality thenF is freely generated.

Recall that for any Boolean algebra (BA) A, the cellularity of A is c(A) = sup{|X| : X is a pairwise-disjoint subset of A}. A pseudo-tree is a partially ordered set (T, ≤) such that for every t in T, the set {r ∊ T : r ≤ t} is a linear order. The pseudo-tree algebra on T, denoted Treealg(T), is the subalgebra of ℘(T) generated by the cones {r ∊ T : r ≥ t}, for t in (...) T. We characterize the cellularity of pseudo-tree algebras in terms of cardinal functions on the underlying pseudo-trees. For T a pseudo-tree, c(Treealg(T)) is the maximum of four cardinals c\sbT, ι\sbT, φ\sbT, and μ\sbT : roughly, c\sbT measures the "tallness" of the pseudo-tree T; ι\sbT the "breadth"; φ\sbT the number of "finite branchings"; and μ\sbT the number of places where T "does not branch." We give examples to demonstrate that all four of these cardinals are needed. (shrink)

We construct Boolean algebras with prescribed behaviour concerning depth for the free product of two Boolean algebras over a third, in ZFC using pcf; assuming squares we get results on ultraproducts. We also deal with the family of cardinalities and topological density of homomorphic images of Boolean algebras (you can translate it to topology - on the cardinalities of closed subspaces); and lastly we deal with inequalities between cardinal invariants, mainly.

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 from [6]. 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 variation on Foreman and Woodin's model in which GCH fails everywhere. (shrink)

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly ϭ-filtered Boolean algebras. We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > N1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free. Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic (...) Boolean algebras of size κ that are tightly ϭ-filtered and c.c.c. These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2<κ isomorphism types of projective Boolean algebras of size κ. (shrink)

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal (...) family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (shrink)

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability on B (where B and C are Boolean algebras, and $\mathscr{B} \subseteq \mathscr{C}$ ) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be (...) extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L. (shrink)

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra.

In Plato's Parmenides we find on the one hand that the One is denied every property , and on the other hand that the One is attributed every property . In the course of the history of Platonism , these assertions - probably meant by Plato as ontological statements of an entirely formal nature - were repeatedly made the starting points of metaphysical speculations. In the Mystical Theology of the Pseudo-Dionysius they became principles of Christian mysticism and negative theology. I (...) shall show that the two assertions can each be interpreted within the ontological framework of ancient and medieval logic in such a manner that it becomes true, and I shall make plausible that they were understood, with regard to their logical core, by pagan and Christian Platonic metaphysicians just as is indicated by that interpretation. The mentioned ontological framework is basically the Boolean algebra of first-order properties. The main points of the interpretation are on the one hand the identification of the One with the maximal element of the algebra of properties, and on the other hand two alternative intuitively prominent mereological definitions of ontic predication. (shrink)

A Boolean algebra B is said to be openly generated if {A: A ≤rc B, |A| = ℵ0} includes a club subset of [ B]ℵ0 . We show: (V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1). (MA+(σ-closed)). Every L∞ℵa -free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using (...) this characterization we also prove the independence of problem 7 in Scepin [15] (Proposition 4.3 and Theorem 4.4). (shrink)

In the book Cardinal Invariants on Boolean Algebras by J. Donald Monk many such cardinal functions are defined and studied. Among them several are generalizations of well known cardinal characteristics of the continuum. Alongside a long list of open problems is given. Focusing on half a dozen of those cardinal invariants some of those problems are given an answer here, which in most of the cases is a definitive one. Most of them can be divided in two groups. The problems (...) of the first group ask about the change on those cardinal functions when going from a given infinite Boolean algebra to its simple extensions, while in the second group the comparison is between a couple of given infinite Boolean algebras and their free product. (shrink)

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that . We prove consistency of the statement “there is a Boolean algebra such that ” and we force a superatomic Boolean algebra such that , and . Next we force a superatomic algebra such that and a superatomic algebra such that . Finally we show that consistently there is (...) a Boolean algebra of size λ such that there is no free sequence in of length λ, there is an ultrafilter of tightness λ and. (shrink)

Recall that a subset X of a Boolean algebra A is independent if for any two finite disjoint subsets F , G of X we have ∏ x∈F x ∏ y∈G −y≠0. The independence of a BA A , denoted by Ind, is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant i , well known in the case A= (...) P / fin . In this article we consider the collection of all cardinalities of infinite maximal independent subsets of a BA A ; we call this set the spectrum of infinite maximal independent subsets , denoted by Spind. Note that infinite maximal independent subsets exist in any BA which is not superatomic. The main result is that any set of infinite cardinals can occur as Spind for some infinite BA A . Beyond this we give results concerning the way that Spind changes under various algebraic operations. However, the basic components of most algebras that we deal with are free algebras. (shrink)

Given an admissible indexing φ of the countable atomless Boolean algebra B, an automorphism F of B is said to be recursively presented (relative to φ) if there exists a recursive function $p \in \operatorname{Sym}(\omega)$ such that F ⚬ φ = φ ⚬ p. Our key result on recursiveness: Both the subset of $\operatorname{Aut}(\mathscr{B})$ consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all "totally nonrecursive" automorphisms, are uncountable. This (...) arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of f and f̄, where f is a permutation of some free basis of B and f̄ is the automorphism of B induced by f. (2) An explicit description of the permutations of ω whose conjugacy classes in $\operatorname{Sym}(\omega)$ are (a) uncountable, (b) countably infinite, and (c) finite. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We (...) include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive (...) algebras determine Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We (...) include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

This paper studies unbounded pseudo-algebraically closed fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.