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 class of theories wider than pe rmutative theories of Mal'cev (1971), for which he gave decision resul ts. 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 SIGMA1-fragment of the theory of T(F)/=E is decidabl e when E is a compact set of axioms. In particular, the existential fr agment of the theory of associative-commutative function symbols is de cidable.