Abstract
We provide a new proof of the following Palasinska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are $${\mathcal{Q}}$$ Q -relation formulas for a protoalgebraic equality free quasivariety $${\mathcal{Q}}$$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for $${\mathcal{Q}}$$ Q when it has definable principal $${\mathcal{Q}}$$ Q -subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.
Highlights
In abstract algebraic logic the following theorem of Katarzyna Palasinska is remarkable [15]: Every protoalgebraic and filter distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many inference rules and axioms
For this purpose we apply the technique of definable principal Q-subrelations. This is equality free quasivariety counterpart of the definable principal subcongruences technique invented by Kirby Baker and Ju Wang [2]
To finish the introduction let us add that the novelty of this paper lies mainly in introducing the proper notion of Q-relation formula for protoalgebraic equality free quasivarieties
Summary
In abstract algebraic logic the following theorem of Katarzyna Palasinska is remarkable [15]: Every protoalgebraic and filter distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many inference rules and axioms By reformulating it into the context of equality free quasivarieties we have (see Section 2 for definitions). To finish the introduction let us add that the novelty of this paper lies mainly in introducing the proper notion of Q-relation formula for protoalgebraic equality free quasivarieties With this tool in hand the results are obtained by translating the arguments from [2] and [21]
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