Term satisfiability in FL${}_{\mathrm{ew}}$-algebras

Abstract

FL${}_{\mathrm{ew}}$-algebras form the algebraic semantics of the full Lambek calculus with exchange and weakening. We investigate two relations, called satisfiability and positive satisfiability, between FL${}_{\mathrm{ew}}$-terms and FL${}_{\mathrm{ew}}$-algebras. For each FL${}_{\mathrm{ew}}$-algebra, the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory; we identify and investigate the complements as fragments of its universal theory. We offer characterizations of those algebras that (positively) satisfy just those terms that are satisfiable in the two-element Boolean algebra providing its semantics to classical propositional logic. In case of positive satisfiability, these algebras are just the nontrivial weakly contractive FL${}_{\mathrm{ew}}$-algebras. In case of satisfiability, we give a characterization by means of another property of the algebra, the existence of a two-element congruence. Further, we argue that (positive) satisfiability problems in FL${}_{\mathrm{ew}}$-algebras are computationally hard. Some previous results in the area of term satisfiability in MV-algebras or BL-algebras are thus brought to a common footing with known facts on satisfiability in Heyting algebras.

the revised version, which benefits from the comments of a reviewer for Theoretical Computer Science, corrects a few minor errors, some parts are reorganized for clarity, and Theorem 5.1 is slightly stronger than in the original version