On <Emphasis Type="Italic">α</Emphasis>-complete retracts of specker lattice ordered groups

Let α be a cardinal. The notion of α-complete retract of a Boolean algebra has been studied by Dwinger. Specker lattice ordered groups were investigated by Conrad and Darnel. Assume that G is a Specker lattice ordered group generated by a Boolean algebra B(G). The notion of α-complete retract of G can be defined analogously as in the case of Boolean algebras. In the present paper we deal with the relations between α-complete retracts of G and α-complete retracts of B(G).     

Initial Chain Algebras on Pseudotrees

We investigate a new class of Boolean algebra, called initial chain algebras on pseudotrees. We discuss the relationship between this class and other classes of Boolean algebras. Every interval algebra, and hence every countable Boolean algebra, is an initial chain algebra. Every initial chain algebra on a tree is a superatomic Boolean algebra, and every initial chain algebra on a pseudotree is a minimally-generated Boolean algebra.We show that a free product of two infinite Boolean algebras is an initial chain algebra if and only if both factors are countable.     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

Downward closure of depth in countable Boolean algebras

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.     

Recursive Homogeneous Boolean Algebras

Isomorphism types of countable homogeneous Boolean algebras are described in [1], in which too is settled the question of whether such algebras are decidable. Precisely, a countable homogeneous Boolean algebra has a decidable presentation iff the set by which an isomorphism type of that algebra is characterized belongs to a class of the arithmetic hierarchy. The problem of obtaining a characterization for homogeneous Boolean algebras which have a recursive presentation remained open. Partially, here we resolve this problem, viz., estimate an exact upper and an exact lower bounds for the set which an isomorphism type of such any algebra is characterized by in terms of the Feiner hierarchy.     

On tightly κ-filtered Boolean algebras

In this article we study the notion of tight ?-filteredness of a Boolean algebra for infinite regular cardinals ?. Tight à₀ \aleph_0 -filteredness is projectivity. We give characterizations of tightly ?-filtered Boolean algebras which generalize the internal characterizations of projectivity given by Haydon, Šcepin, and Koppelberg (see [15] or [17]). We show that for each ? there is an rc-filtered Boolean algebra which is not tightly ?-filtered. This generalizes a result of Šcepin (see [15]). We prove that no complete Boolean algebra of size larger than à₂ \aleph_2 is tightly à₁ \aleph_1 -filtered. We give a new example of a model of set theory where \frak P(w) \frak P(\omega) is tightly s-filtered. We study the effect of the tight s-filteredness of \frak P(w) \frak P(\omega) on the automorphism group of \frak P(w)/fin \frak P(\omega)/fin .     

A Note on Boolean Algebras with Few Partitions Modulo some Filter

We show that for every uncountable regular κ and every κ-complete Boolean algebra B of density ≤ κ there is a filter F ? B such that the number of partitions of length < modulo κF is ≤2^<κ. We apply this to Boolean algebras of the form P(X)/I, where I is a κ-complete κ-dense ideal on X. Mathematics Subject Classification: 06E05, 03C20.     

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.     

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.     

Intuitionistic choice and classical logic

Abstract. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of the Boolean algebra we can directly extract effective content from -statements true in the model. All the arguments of the present paper can be formalised in Martin-L?f's constructive type theory with generalised inductive definitions. Received: 20 March 1997 / Revised version: 20 February 1998     

BOOLEAN ALGEBRAS IN AST

In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory (AST). 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.     

Boundedness of characters in locally a-convex algebras

We characterize advertibly complete topological algebras among those whose completions are algebras. We show that the set of non-zero characters in a commutative, advertibly complete,M-complete locallyA-convex algebra is equibounded whenever every element of the algebra is bounded.     

Decomposition of vector-valued additive functions

From the additive nonnegative fonctions, defined on a - algebra of subsets, the property of countable additivity separates a class of regular objects, namely measures. Among additive Banach-valued functions, countable additivity is already not necessary for non-pathology. In the paper one isolates a class of regular vector-valued additive functions (measures) and one proves a theorem on the decomposition of a vector-valued additive function into the sum of a regular component (measure) and a purely pathological additive function, similar to a purely finitely additive measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 148–156, 1989.     

Theorem on the minorant. The denumerability of Maharam's problem

It is proved that on a normed denumerably complete Boolean algebra each continuous exterior measure majorizes at least one measure. With the aid of this result and of the Kelley numbers, Maharam's well-known problem on the seminormability and normability of Boolean algebras is transformed to a problem on the additive minorants of semimeasures continuous from one side on denumerable algebras.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 597–604, May, 1977.     

Some Boolean Algebras with Finitely Many Distinguished Ideals I

We consider 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 where the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th_prin and Th_sa. If T is a theory in a first order language and α is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(α) be the interval algebra of α. Using rank diagrams, we show that Sentalg(Th_prin) ? intalg(ω⁴), Sentalg(Th_max) ? intalg(ω³) ? Sentalg(Th_ac), and Sentalg(Th_sa) ? intalg(ω² + ω²). For Th_max and Th_ac we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic.     

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.     

Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Internal injectivity of Boolean algebras in MSet

This paper deals with the internal notion of injectivity for Boolean algebras in the topos of M-sets. Given that, for ordinary Boolean algebraas, injectivity is the same as completeness (Sikorski's theorem) and the injective hull is the same as normal completion, we investigate here how the internal notion of completeness relates to internal injectivity. Further, we consider the internal injectivity of the initial Boolean algebra 2 which is equivalent to the prime ideal theorem for Boolean algebras in this topos. Before we turn specificially to Boolean algebras, we develop the bassic general facts concerning internal injectivity in MSet for arbitrary equational classes of algebras.