Shelah rank for Boolean algebras and some application to elementary theories I

1980 ◽  
Vol 10 (1) ◽  
pp. 247-257 ◽  
Author(s):  
Piero Mangani ◽  
Annalisa Marcja
Keyword(s):  
Boolean Algebras ◽  
Elementary Theories

A remark on Martin's Conjecture

2001 ◽  
Vol 66 (1) ◽  
pp. 401-406
Author(s):  
Su Gao
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
First Order ◽  
First Order Theory ◽  
Elementary Theories

AbstractWe prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

scholarly journals Alfred Tarski and undecidable theories

1986 ◽  
Vol 51 (4) ◽  
pp. 890-898 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Elementary Theory ◽  
Logical Consequence ◽  
Quantifier Elimination ◽  
Equational Theory ◽  
Boolean Algebras ◽  
Decision Problems ◽  
Alfred Tarski ◽  
Elementary Theories

Alfred Tarski identified decidability within various logical formalisms as one of the principal themes for investigation in mathematical logic. This is evident already in the focus of the seminar he organized in Warsaw in 1926. Over the ensuing fifty-five years, Tarski put forth a steady stream of theorems concerning decidability, many with far-reaching consequences. Just as the work of the 1926 seminar reflected Tarski's profound early interest in decidability, so does his last work, A formalization of set theory without variables, a monograph written in collaboration with S. Givant [8−m]. An account of the Warsaw seminar can be found in Vaught [1986].Tarski's work on decidability falls into four broad areas: elementary theories which are decidable, elementary theories which are undecidable, the undecidability of theories of various restricted kinds, and what might be called decision problems of the second degree. An account of Tarski's work with decidable elementary theories can be found in Doner and van den Dries [1987] and in Monk [1986] (for Boolean algebras). Vaught [1986] discusses Tarski's contributions to the method of quantifier elimination. Our principal concern here is Tarski's work in the remaining three areas.We will say that a set of elementary sentences is a theory provided it is closed with respect to logical consequence and we will say that a theory is decidable or undecidable depending on whether it is a recursive or nonrecursive set. The notion of a theory may be restricted in a number of interesting ways. For example, an equational theory is just the set of all universal sentences, belonging to some elementary theory, whose quantifier-free parts are equations between terms.

scholarly journals Arities and Aritizabilities of First-Order Theories

2021 ◽  
Author(s):  
Sergey Vladimirovich Sudoplatov
Keyword(s):  
Boolean Algebras ◽  
First Order ◽  
Elementary Theories

We study and describe possibilities for arities of elementary theories and of their expansions. Links for arities with respect to Boolean algebras, to disjoint unions and to compositions of structures are shown. Arities and aritizabilities are semantically characterized. The dynamics for arities of theories is described.

Applications of Finite Models Properties in Approximation and Algorithmic Logics

1991 ◽  
Vol 14 (1) ◽  
pp. 91-108
Author(s):  
Jarosław Stepaniuk
Keyword(s):  
Finite Models ◽  
Elementary Theories

The purpose of this paper is to investigate some aspects concerning elementary theories of finite models and to give the applications in approximation logics and algorithmic theory of dictionaries.

Information storage and retrieval systems with incomplete information I

1979 ◽  
Vol 2 (1) ◽  
pp. 17-41
Author(s):  
Michał Jaegermann
Keyword(s):  
Incomplete Information ◽  
Information Storage And Retrieval ◽  
Boolean Algebras ◽  
Information Storage ◽  
Storage And Retrieval ◽  
Retrieval Systems ◽  
Theory Of Information

In the paper is developed a theory of information storage and retrieval systems which arise in situations when a whole possessed information amounts to a fact that a given document has some feature from properly chosen set. Such systems are described as suitable maps from descriptor algebras into sets of subsets of sets of documents. Since descriptor algebras turn out to be pseudo-Boolean algebras, hence an “inner logic” of our systems is intuitionistic. In the paper is given a construction of systems and are considered theirs properties. We will show also (in Part II) a formalized theory of such systems.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

scholarly journals Handbook of boolean algebras

1990 ◽  
Vol 84 (1) ◽  
pp. 136
Author(s):  
Gian-Carlo Rota
Keyword(s):  
Boolean Algebras
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