Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska's Theorem Revisited

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are Q-relation formulas for a protoalgebraic equality free quasivariety 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 Q when it has definable principal 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.