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.
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Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko
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Hoehnke, HJ. Quasi-varieties: A special access. Stud Logica 78, 249–260 (2004). https://doi.org/10.1007/s11225-005-0260-z
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DOI: https://doi.org/10.1007/s11225-005-0260-z