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Decidable properties of finite sets of equations in trivial languages

Published online by Cambridge University Press:  12 March 2014

Cornelia Kalfa
Cornelia Kalfa*
Affiliation:
Aristotle University, Salonica, Greece
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1333 - 1338
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[1]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[2]Ehrenfeucht, A., Decidability at the theory of one function, Notices of the American Mathematical Society, vol. 6 (1959), p. 268.Google Scholar
[3]Kalfa, C., Decision problems concerning sets of equations, Ph.D. Thesis, University of London, London, 1980.Google Scholar
[4]Kalfa, C., Decision problems concerning properties of finite sets of equations (to appear).Google Scholar