Quasi-Nelson Algebras

Abstract We introduce a generalization of Nelson algebras having a not-necessarily involutive negation; we suggest to dub this class quasi-Nelson algebras in analogy with quasi-De Morgan lattices, these being a non-involutive generalization of De Morgan lattices. We show that, similarly to the involutive case (and perhaps surprisingly), our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special products of Heyting algebras, which generalize the well-known construction for representing algebraic models of Nelson's constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic viewed as a conservative expansion of the negation-free fragment of intuitionistic logic. The equivalence of the three presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact may be the possibility of opening new ways to (i) obtain deeper insights into the distinguishing feature of Nelson's logic (the Nelson axiom) and its algebraic counterpart; (ii) be able to investigate certain purely algebraic properties (such as 3-potency and (0,1)-congruence orderability) in a more general setting

Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions

Involutive Commutative Residuated Lattice without Unit: Logics and Decidability

We investigate involutive commutative residuated lattices without unit, which are commutative residuated lattice-ordered semigroups enriched with a unary involutive negation operator. The logic of this structure is discussed and the Genzten-style sequent calculus of it is presented. Moreover, we prove the decidability of this logic.Comment: 16 page

Paraconsistent Modal Logics

AbstractWe introduce a modal expansion of paraconsistent Nelson logic that is also as a generalization of the Belnapian modal logic recently introduced by Odintsov and Wansing. We prove algebraic completeness theorems for both logics, defining and axiomatizing the corresponding algebraic semantics. We provide a representation for these algebras in terms of twist-structures, generalizing a known result on the representation of the algebraic counterpart of paraconsistent Nelson logic

Powerset residuated algebras

We present an algebraic approach to canonical embeddings of arbitrary residuated algebras into powerset residuated algebras. We propose some construction of powerset residuated algebras and prove a representation theorem for symmetric residuated algebras