On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

The overlap algebra of regular opens

Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, overlap algebras offer a negation-free framework suitable, among other things, for the development of point-free topology. A lot of topology can be done “inside” the language of overlap algebra. In particular, we prove that the collection of all regular open subsets of a topological space is an example of overlap algebra which, under natural hypotheses, is atomless. Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful both in constructive mathematics and for applications in computer science.     

Computably categorical Boolean algebras enriched by ideals and atoms

We describe computably categorical Boolean algebras whose language is enriched by one-place predicates that distinguish a finite set of ideals and atoms with respect to some ideals in this set.     

The Nikodym property and cardinal characteristics of the continuum

We present a general method of constructing Boolean algebras with the Nikodym property and of some given cardinalities. The construction is dependent on the values of some classical cardinal characteristics of the continuum. As a result we obtain a consistent example of an infinite Boolean algebra with the Nikodym property and of cardinality strictly less than the continuum $\mathfrak{c}$. It follows that the existence of such an algebra is undecidable by the usual axioms of set theory. Besides, our results shed some new light on the Efimov problem and cofinalities of Boolean algebras.     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.     

Greechie diagrams of orthomodular partial algebras

Greechie diagrams are well known graphical representations of orthomodular partial algebras, orthomodular posets and orthomodular lattices. For each hypergraph D a partial algebra ⟦D⟧ = (A; ⊕, ′, 0) of type (2,1,0) can be defined. A Greechie diagram can be seen as a special hypergraph: different points of the hypergraph have different interpretations in the corresponding partial algebra ⟦D⟧, and each line in the hypergraph has a maximal Boolean subalgebra as interpretation, in which the points are the atoms. This paper gives some generalisations of the characterisations in [K83] and [D84] of diagrams which represent orthomodular partial algebras (= OMAs), and we give an algorithm how to check whether a given hypergraph D is an OMA-diagram whose maximal Boolean subalgebras are induced by the lines of the hypergraph. Received July 22, 2004; accepted in final form February 1, 2007.     

Condition for inducing topology in a vector lattice by the order completion

Properties of an order topology in vector lattices and Boolean algebras are studied. The main result is the following: in a vector lattice or a Boolean algebra with the condition of “closure by one step” (a generalization of the well-known “regularity” property of Boolean algebras and K-spaces) the order topology is induced by the topology of its Dedekind completion. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 204–207.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

Generalizations of Boolean products for lattice-ordered algebras

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL_w-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.     

Cardinalities of σ-complete OML's

In this paper we show that the Comfort-Hager result on cardinalities of-complete Boolean algebras is also true for-complete OML's having a bound on the number of complements. Using the Kaplansky theorem on continuous geometries we get a result on modular ortho-lattices. We also get a result on cardinalities of saturated OML's.Presented by R. McKenzie.Supported by the NSF of Srbija through Math. Inst., Grant #401A.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Boolean inequations

In this paper we consider Boolean inequations i.e. the inequations of the form f(X)≠0, where f is a Boolean function. The basic idea in this paper is: the inequation f(X)≠0 means that there exists p such that f(X)=p and p≠0. We give the formula which determines all the solutions of Boolean inequation.     

Σ-bounded algebraic systems and universal functions. II

Ershov algebras, Boolean algebras, and abelian p-groups are Σ-bounded systems, and there exist universal Σ-functions in hereditarily finite admissible sets over them.     

More constructions for Boolean algebras

 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 . Received: 9 September 1998 / Published online: 7 May 2002     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Computable Homogeneous Boolean Algebras and a Metatheorem

We consider computable homogeneous Boolean algebras. Previously, countable homogeneous Boolean algebras have been described up to isomorphism and a simple criterion has been found for the existence of a strongly constructive (decidable) isomorphic copy for such. We propose a natural criterion for the existence of a constructive (computable) isomorphic copy. For this, a new hierarchy of -computable functions and sets is introduced, which is more delicate than Feiner's. Also, a metatheorem is proved connecting computable Boolean algebras and their hyperarithmetical quotient algebras.