Dense Elements and Classes of Residuated Lattices

In this paper we study the dense elements and the radical of a residuated lattice, residuated lattices with lifting Boolean center, simple, local, semilocal and quasi-local residuated lattices. BL-algebras have lifting Boolean center; moreover, Glivenko residuated lattices which fulfill a certain equation (that is satisfied by BL-algebras) have lifting Boolean center.Comment: 16 page

Term satisfiability in FLew_\mathrm{ew}-algebras

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.Comment: 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 versio

Semisimplicity, Glivenko theorems, and the excluded middle

We formulate a general, signature-independent form of the law of the excluded middle and prove that a logic is semisimple if and only if it enjoys this law, provided that it satisfies a weak form of the so-called inconsistency lemma of Raftery. We then show that this equivalence can be used to provide simple syntactic proofs of the theorems of Kowalski and Kracht characterizing the semisimple varieties of FLew-algebras and Boolean algebras with operators, and to extend them to FLe-algebras and Heyting algebras with operators. Moreover, under stronger assumptions this correspondence works at the level of individual models: the semisimple models of such a logic are precisely those which satisfy an axiomatic form of the law of the excluded middle, and a Glivenko-like connection obtains between the logic and its extension by the axiom of the excluded middle. This in particular subsumes the well-known Glivenko theorems relating intuitionistic and classical logic and the modal logics S4 and S5. As a consequence, we also obtain a description of the subclassical substructural logics which are Glivenko related to classical logic.Comment: 47 pages, 0 figure