Manuela Busaniche

Constructive Logic with Strong Negation as a Substructural Logic

Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic. We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN. For instance, we prove that Nilpotent minimum logic is the extension of CLSN by the prelinearity axiom. This generalizes the well-known result by Monteiro and Vakarelov that three-valued Łukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued Łukasiewicz logic is proved.

Free nilpotent minimum algebras

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Spectral Duality for Finitely Generated Nilpotent Minimum Algebras, with Applications

In nilpotent minimum logic, the conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. We shall focus attention on finitely axiomatized theories in nilpotent minimum logic; their algebraic counterparts are finite NM-algebras. Building on results in [1] and [2], we establish a spectral (or Stone-type) duality for finite NM-algebras. We describe the dual of a finite NM-algebra as a forest (i.e., a finite partially ordered set such that below any element there lies a subset that inherits a total order), such that each one of its trees (i.e., partially ordered subsets with a minimum) is enriched by one additional bit of information. Homomorphisms of NM-algebras dualise to order-preserving maps (between the corresponding forests) satisfying appropriate additional conditions. This seems to be the first instance of a (finite) spectral duality for a logic that is based on a discontinuous t-norm. We then show that the construction is actually useful in obtaining further results: an explicit description of finite coproducts of finite NM-algebras; a strong form of amalgamation for finite NM-algebras, along with the strongest possible form of the Deduction Theorem for nilpotent minimum logic; a functional representation of free finitely generated NM-algebras; an exact recursive formula for the cardinality of free finitely generated NM-algebras.

Residuated Lattices as an Algebraic Semantics for Paraconsistent Nelson's Logic

The class of NPc-lattices is introduced as a quasivariety of commutative residuated lattices, and it is shown that the class of pairs (A,A+) such that A is an NPc-lattice and A+ is its positive cone, is a matrix semantics for Nelson paraconsistent logic.

Bouligand-Severi tangents in MV-algebras.

In their recent seminal paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.

Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups

Abstract. A unital

Polyhedral MV-algebras

\ell

Completions in Subvarieties of BL-Algebras

-group (G,u)

A Categorical Equivalence for Stonean Residuated Lattices

(G,u)

Poset Product and BL-Chains

is an abelian group G