A sound and complete set of axioms and inference rules for quasi-equational logic

Question

Equational logic has some axioms and inference rules to derive equations from other equations. What about quasi-equational logic? Is there, in some text, a set of sound and complete axioms and inference rules for quasi-equational logic, along with the proof that the set is indeed sound and complete?

Answer 1

In Partial Horn logic and cartesian categories, Palmgren and Vickers give a presentation of partial Horn logic, which is equivalent to quasi-equational logic, and prove it sound and complete.