Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantiier projectivity, using a semantic treatment that regards quantiiers as models rather than variables, and regards valuations as model morphisms rather than functions. Applications include many and order sorted conditional] equational logics, Horn clause lo...

Equational logic programming is an extended programming paradigm of equational programming. Central to the notion of equational logic programming is the problem of solving equations, which is also called unification in equational theories. In this paper, we investigate the problem of solving equations in O’Donnell’s equational language. We define an equality theory for this language which adequ...

We give a general method for constructing lattices L whose equational theories are inherently nonfinitely based. This means that the equational class (that is, the variety) generated by L is locally finite and that L belongs to no locally finite finitely axiomatizable equational class. We also provide an example of a lattice which fails to be inherently nonfinitely based but whose equational th...

We show that the generalized variant of formal systems where the underlying equational specifications are membership equational theories, and where the rules are conditional and can have equations, memberships and rewrites in the conditions is reflective. We also show that membership equational logic, many-sorted equational logic, and Horn logic with equality are likewise reflective. These resu...

Birkhoff’s completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneo...

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by parame...