QUANTIFIERS ON LATTICES WITH AN ANTITONE INVOLUTION
Ivan Chajda, Helmut Länger · Demonstratio Mathematica · 2009
Quantifiers on lattices with an antitone involution are considered and it is proved that the poset of existential quantifiers is antiisomorphic to the poset of relatively complete sublattices.So-called monadic algebras are often used as an algebraic axiomatization of predicate calculus, see e. g. [1] -[4].In fact, every one of these predicate logics is a lattice with respect to the induced order.Monadic algebras were investigated in connection with lattices representing many-valued predicate calculus by Rutledge ([4]), MV-algebras (which are the algebraic counterpart of many-valued logics) by Di Nola and Grigolia ([2]), basic algebras (which are generalizations of MV-algebras) by Chajda and Kolarik ([1]) and residuated lattices by Rachunek and Svrcek ([3]).Existential quantifiers were already characterized by means of algebraic methods (using closure operators and relatively complete subalgebras) for MV-algebras, pseudo MV-algebras and a number of other algebras used in the axiomatization of both classical and non-classical (in particular many-valued) logics (including basic algebras).Since all these algebras are bounded lattices the natural question arises if a general and unified approach can be developed.The aim of the present paper is to show that quantifiers may be characterized for all possible logics having a bounded lattice as their underlying structure.DEFINITION 1.A lattice with an antitone involution is an algebra (L, V, A, -i) of type (2,2,1) such that (L, V, A) is a lattice and 2000 Mathematics Subject Classification: 03G10, 06C15.