On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Some Boolean algebras with finitely many distinguished ideals II

We describe the countably saturated models and prime models (up to isomorphism) of 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 such that the supremum of the ideal exists, and the theory Th_sa of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.     

Categoricity in hyperarithmetical degrees

We obtain, under certain assumptions, necessary and sufficient conditions for a recursive structure to be Δ⁰_α-categorical. This is done using the author's α-systems to construct suitable Δ⁰_α+1 functions. We show how these results may be applied, for example, to superatomic Boolean algebras.     

Torsion classes of generalized Boolean algebras

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T _g and R _g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T _g and R _g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T _g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.     

Categoricity results for L∞κ

Let V denote a variety of algebras in a countable language. An algebra is said to be L_∞κ-free if it is L_∞κ-equivalent to a (V-) free algebra. If every L_∞ω1-free algebra of cardinality ω₁ is free, then for all infinite cardinals κ every L_∞κ-free algebra of cardinality κ is free. Further, assuming suitable set-theoretic hypotheses, if there is a non-free L_∞ω1-free algebra of cardinality ω₁, then for a proper class of cardinals κ there are non-free L_∞κ-free algebras of cardinality κ. The varieties in which the class of free algebras are definable by a sentence in L_ω1ω are characterized as the weak Schreier varieties in which every L_∞ω-free algebra of cardinality ω₁ is free. A weak Schreier variety is one in which every L_∞ω-elementary substructure of a free algebra is free. In fact, assuming suitable set-theortic hypotheses, for weak Schreier varieties the class of free algebras is definable in L_∞∞ iff it is definable in L_ω1ω. Varieties in uncountable languages are also considered.     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

Decomposable <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>-valued measures as measurable bundles

Decomposable L ₀-valued measures on a complete Boolean algebra B are considered. Criterion for B to be representable as a measurable bundle of continuous (respectively, atomic) Boolean algebras is given.     

The structure of separable Dynkin algebras

Abstract Dynkin algebras are studied. Such algebras form a useful instrument for discussing probabilities in a rather natural context. Abstractness means the absence of a set-theoretic structure of elements in such algebras. A large useful class of abstract algebras, separable Dynkin algebras, is introduced, and the simplest example of a nonseparable algebra is given. Separability allows us to define appropriate variants of Boolean versions of the intersection and union operations on elements. In general, such operations are defined only partially. Some properties of separable algebras are proved and used to obtain the standard intersection and union properties, including associativity and distributivity, in the case where the corresponding operations are applicable. The established facts make it possible to define Boolean subalgebras in a separable Dynkin algebra and check the coincidence of the introduced version of the definition with the usual one. Finally, the main result about the structure of separable Dynkin algebras is formulated and proved: such algebras are represented as set-theoretic unions of maximal Boolean subalgebras. After preliminary preparation, the proof reduces to the application of Zorn’s lemma by the standard scheme.     

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.     

On a type of distributivity of lattice ordered groups

Let m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r _m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.     

How to construct extensional combinatory algebras

We develop a slight modification of Engeler's graph algebras, yielding extensional combinatory algebras. It is shown that by this construction we get precisely the class of Scott's D_∞-models generated by complete atomic Boolean algebras. In section 3 we construct extensional substructures of graph-algebras and Pω-models.     

on the o-minimal LS-category

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture.     

ŁΠ logic with fixed points

We study a system, μ?Π, obtained by an expansion of ?Π logic with fixed points connectives. The first main result of the paper is that μ?Π is standard complete, i.e., complete with regard to the unit interval of real numbers endowed with a suitable structure. We also prove that the class of algebras which forms algebraic semantics for this logic is generated, as a variety, by its linearly ordered members and that they are precisely the interval algebras of real closed fields. This correspondence is extended to a categorical equivalence between the whole category of those algebras and another category naturally arising from real closed fields. Finally, we show that this logic enjoys implicative interpolation.     

Expansions of ordered fields without definable gaps

In this paper we are concerned with definably, with or without parameters, (Dedekind) complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o‐minimal if and only if all their definable subsets are finite disjoint unions of definably connected (definable) subsets. We pay attention to how simply (in terms of the quantifier complexity and/or usage of parameters) a definable gap in an expansion is so. Next we prove that over parametrically definably complete expansions of ordered fields, all one‐to‐one definable (with parameters) continuous functions are monotone and open. Moreover, in both parameter and parameter‐free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine‐Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets.     

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.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.