Boolean sentence algebras: Isomorphism constructions

AbstractAssociated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions.

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Decidable properties of finite sets of equations in trivial languages

Journal of Symbolic Logic ◽

10.2307/2274282 ◽

1984 ◽

Vol 49 (4) ◽

pp. 1333-1338

Author(s):

Cornelia Kalfa

Keyword(s):

Equational Theory ◽

Order Theory ◽

Operation Symbol ◽

First Order ◽

First Order Theory ◽

Order Language ◽

Finite Set ◽

Joint Embedding ◽

Algebraic Language ◽

Joint Embedding Property

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):P0(Σ) = the equational theory of Σ is equationally complete.P1(Σ) = the first-order theory of Σ is complete.P2(Σ) = the first-order theory of Σ is model-complete.P3(Σ) = the first-order theory of the infinite models of Σ is complete.P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.P5(Σ) = Σ has the joint embedding property.In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

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Decision problems concerning properties of finite sets of equations

Journal of Symbolic Logic ◽

10.2307/2273945 ◽

1986 ◽

Vol 51 (1) ◽

pp. 79-87 ◽

Cited By ~ 2

Author(s):

Cornelia Kalfa

Keyword(s):

Order Theory ◽

List Type ◽

Finite Sets ◽

Operation Symbol ◽

First Order ◽

First Order Theory ◽

Rank One ◽

Joint Embedding ◽

Algebraic Language ◽

Joint Embedding Property

In this paper a general method of proving the undecidability of a property P, for finite sets Σ of equations of a countable algebraic language, is presented. The method is subsequently applied to establish the undecidability of the following properties, in almost all nontrivial such languages:1. The first-order theory generated by the infinite models of Σ is complete.2. The first-order theory generated by the infinite models of Σ is model-complete.3. Σ has the joint-embedding property.4. The first-order theory generated by the models of Σ with more than one element has the joint-embedding property.5. The first-order theory generated by the infinite models of Σ has the joint-embedding property.A countable algebraic language ℒ is a first-order language with equality, with countably many nonlogical symbols but without relation symbols, ℒ is trivial if it has at most one operation symbol, and this is of rank one. Otherwise, ℒ is nontrivial. An ℒ-equation is a sentence of the form , where φ and ψ are ℒ-terms. The set of ℒ-equations is denoted by Eqℒ. A set of sentences is said to have the joint-embedding property if any two models of it are embeddable in a third model of it.If P is a property of sets of ℒ-equations, the decision problem of P for finite sets of ℒ-equations is the problem of the existence or not of an algorithm for deciding whether, given a finite Σ ⊂ Eqℒ, Σ has P or not.

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The elementary class of products of totally ordered abelian group

Journal of Symbolic Logic ◽

10.2307/2274920 ◽

1991 ◽

Vol 56 (1) ◽

pp. 295-299 ◽

Cited By ~ 1

Author(s):

Daniel Gluschankof

Keyword(s):

Abelian Groups ◽

Algebraic Theory ◽

Unary Operation ◽

Order Theory ◽

Direct Products ◽

First Order ◽

First Order Theory ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

General Goal

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

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