Abstract
We show that T( F)/= E can be completely axiomatized when = E is a quasi-free theory. Quasi-free theories are a class of theories wider than permutative theories of Mal'cev (1971), for which he gave decision results. As an example of application, we show that the first-order theory of T( F)/= E is decidable when E is a set of ground equations. Besides, we prove that the Σ 1-fragment of the theory of T( F)/= E is decidable when E is a compact set of axioms. In particular, the existential fragment of the theory of associative–commutative function symbols is decidable.
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