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 Í (X:QE)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 (
T Î | Th |\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.
Keywords Varieties and quasi-varieties of partial algebras - partial theories - partial Dale monoids - Mal
cev clones
Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by
M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko