Rotations of Gödel algebras with modal operators

TL;DR

We address algebraic semantics for fuzzy modal logic by classifying directly indecomposable NM-algebras with modal operators via rotations of Gödel algebras with operators. The paper extends the connected/disconnected rotation construction to the modal setting, defining $NM^{+}({f A})$ and $NM^{-}({f A})$ and augmenting them with modal operators $\Box$ and $\lozenge$ to obtain NMAO$^{+}$ and NMAO$^{-}$. Two main representation theorems show that, under suitable hypotheses, $(A, \Box, \lozenge)$ with $A$ directly indecomposable is isomorphic to ${\bf G}(NM^{\pm}(A))$ and conversely that a directly indecomposable NM$^+$ or NM$^-$ algebra with operators arises as ${NM}^{\pm}({\bf G(B)})$ for some d.i. NM-algebra ${\bf B}$. The results connect modal NM-algebras to forest-frame duals and point to future work on twist-structures and broader operator frameworks within NM-algebraic semantics.

Abstract

The present paper is devoted to study the effect of connected and disconnected rotations of Gödel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable Gödel algebras endowed with modal operators.

Figures (2)

• Figure 1: A d.i. Gödel algebra with a $\Box$ operator satisfying (N1) (a) and a $\lozenge$ operator (b), and the corresponding d.i. NM$^+$-algebra with the operators $\boxminus$ (c) and $\boxminus$ (d).
• Figure 2: A d.i. Gödel algebra endowed with a $\Box$ operator (a) and a $\lozenge$ operator (b) satisfying (N1), (SM$\Box$) and (SM$\lozenge$), and the corresponding d.i. NM$^-$-algebra with the operators $\boxminus$ (c) and $\mathord{\lozenge\space\footnotesize{}}\,$ (d).

Theorems & Definitions (21)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Theorem 2.6
• Theorem 2.7
• Definition 3.1
• Lemma 3.2
• Definition 3.3