An Alternative Notion of Quantifiers on Three-Valued Łukasiewicz Algebras
Alejandro Petrovich and Marina Lattanzi
The notion of an existential m-quantifier on a three-valued Łukasiewicz algebra is introduced and studied. The class of three-valued Łukasiewicz algebras endowed with an existential m-quantifier is equational and hence determines a variety denoted by ?m. We prove that the existential m-quantifiers are interdefinable with the existential quantifiers introduced by Luiz Monteiro. Hence every algebra in ?m is term equivalent to a monadic three-valued Łukasiewicz algebra. We characterize the simple algebras in the variety ?m which turns out to be semisimple. We also find some connections between existential m-quantifiers and those existential quantifiers defined on bounded distributive lattices considered by Cignoli in , including Boolean algebras. Finally, we prove a Kripke-style representation theorem.
Keywords: Three-valued Łukasiewicz algebras, quantifiers, MV-algebras.