Complete Axiomatizations of Some Quotient Term Algebras

We show that T (F)= = E can be completely axiomatized when = E is a quasi-free theory. Quasi-free theories are a wider class of theories than permutative theories of Mal71] for which Mal'cev gave decision results. As an example of application, we show that the rst order theory of T (F)= = E is decidable when E is a set of ground equations. Besides, we prove that the 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.