Priestley duality for (modal) N4-lattices

N4-lattices are the algebraic semantics of paraconsistent Nelson logic, which was introduced in [1] as an inconsistency-tolerant counterpart of the better-known logic of Nelson [7, 13]. Paraconsistent Nelson logic combines interesting features of intuitionistic, classical and many-valued logics (e.g., Belnap-Dunn four-valued logic); recent work has shown that it can also be seen as one member of the wide family of substructural logics [15]. The work we present here is a contribution towards a better topological understanding of the algebraic counterpart of paraconsistent Nelson logic, namely a variety of involutive lattices called N4-lattices in [8]. A Priestley-style duality for these algebras has already been introduced by Odintsov [10]. The main difference between his approach and ours is that we only rely on Esakia duality for Heyting algebras [4], whereas [10] uses both Esakia duality and the duality for De Morgan algebras [2, 3]; as a consequence, the description of dual spaces that we obtain is, in our opinion, much simpler. Moreover, [10] only deals with N4-lattices whose lattice reduct is bounded, whereas we show that our treatment extends to the non-bounded case as well. We also consider N4-lattices expanded with a monotone modal operator, which have been recently introduced in the algebraic investigation of modal expansions of Belnap-Dunn logic [12, 11, 14]. Building on duality theory for distributive lattices with modal operators [5, 6], we introduce a duality for these modal N4-lattices, which can moreover be employed to provide a neighborhood semantics for the logic of [14].