Abstract

Quasi-equational logic concerns with a completeness theorem, i. e. a list of general syntactical rules such that, being given a set of graded quasi-equations Q, the closure Cl Q = Qeq Fun Q can be derived from $Q \subseteq (X:QE)$ by the given rules. Those rules do exist, because our consideration could be embedded into the logic of first order language. But, we look for special (“quasi-equational”) rules. Suitable rules were already established for the (non-functorial) case of partial algebras in Definition 3.1.2 of [27], p. 108, and [28], p. 102. (For the case of total algebras, see [35].) So, one has to translate these rules to the (functorial) language of partial theories $\underline T \in \left| {\underline {\mathcal{T}h} } \right|$ .Surprisingly enough, partial theories can be replaced up to isomorphisms by partial “Dale” monoids (cf. Section 3), which, in the total case are ordinary monoids