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Decision problems concerning properties of finite sets of equations

Published online by Cambridge University Press:  12 March 2014

Cornelia Kalfa
Cornelia Kalfa*
Affiliation:
Department of Mathematics, Aristotle University, Thessaloniki, Greece
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Extract

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. 1. The first-order theory generated by the infinite models of Σ is complete.

  2. 2. The first-order theory generated by the infinite models of Σ is model-complete.

  3. 3. Σ has the joint-embedding property.

  4. 4. The first-order theory generated by the models of Σ with more than one element has the joint-embedding property.

  5. 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 1 , March 1986 , pp. 79 - 87
Copyright
Copyright © Association for Symbolic Logic 1986

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References

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