Twist-structures isomorphic to modal Nelson lattices

TL;DR

This work addresses the challenge of combining Nelson constructive logic with modal reasoning by using twist-structures over Heyting algebras to support two non-interdefinable modal operators. It develops modal Nelson lattices as twist-product constructions over modal Heyting algebras and proves a categorical equivalence with boolean-filtered pairs (M,F), enabling a robust algebraic and semantic framework. Two subvarieties, modal normal Nelson lattices and modal φ-regular Nelson lattices, are characterized and connected to underlying Heyting structures via explicit representations and regularity conditions. A comprehensive topological duality is then established, extending Esakia duality to modal settings and yielding dual equivalences between the relevant categories of algebras and spaces, thus providing a unified semantic toolkit for the combined logics.

Abstract

In this paper, we introduce a new variety of Heyting algebras with two unary modal operators that are not interdefinable but satisfy the weakest condition necessary to define modal operators on Nelson lattices. To achieve this, we utilize the representation of Nelson lattices as twist structures over Heyting algebras and establish a categorical equivalence. Finally, we develop a topological duality for this new variety and apply it to derive a topological duality for modal Nelson lattices.

Figures (4)

• Figure 1: The Hasse diagram of a modal Heyting algebra $\mathbf{M} = \langle \mathbf{H}, \square, \Diamond \rangle$. The behavior of the $\Diamond$ operator is depicted in red on the right, and the behavior of the $\square$ operator is shown in blue on the left.
• Figure 2: The Hasse diagram of the modal Nelson algebra $\mathbf{N}(\mathbf{M})$. The behavior of the $\Diamond$ operator is depicted in red on the right, and the behavior of its dual $\square$ operator is shown in blue on the left.
• Figure 3: Commutative diagrams for $\mathbf{E}$ and $\mathbf{F}$ where $\mathbf{N}_1=\langle\mathbf{A}_1,\blacksquare_1,\blacklozenge_1\rangle$, $\mathbf{N}_2=\langle\mathbf{A}_2,\blacksquare_2,\blacklozenge_2\rangle$, and $P_1=( \mathbf{M}_1,F_1)$ and $P_2=( \mathbf{M}_2,F_2)$.
• Figure 4: Commutative diagrams for $\mathbf{G}$ and $\mathbf{J}$ where $\mathbf{M}_1=\langle\mathbf{H}_1,\square_1,\Diamond_1\rangle$, $\mathbf{H}_2=\langle\mathbf{H}_2,\square_2,\Diamond_2\rangle$, and $\mathcal{X}_1=\langle X_1,\tau_1,\leq_1,\eta_1,\eta_2\rangle$ and $\mathcal{X}_2=\langle X_2,\tau_2,\leq_2,\eta'_1,\eta'_2\rangle$.

Theorems & Definitions (77)

• Corollary 1
• Theorem 1: Sendlewski + Theorem 3.1 in Busaniche
• Corollary 2
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Theorem 2
• Definition 1