Rotations of G\"odel algebras with modal operators

TL;DR

This paper investigates how connected and disconnected rotations of G"odel algebras with modal operators affect their structure, providing a (quasi-)equational characterization of resulting nilpotent minimum algebras.

Contribution

It introduces a novel framework for understanding rotations of G"odel algebras with modal operators, characterizing the resulting structures as nilpotent minimum algebras.

Findings

Connected and disconnected rotations produce nilpotent minimum algebras with modal operators.

These structures are fully characterized as rotations of directly indecomposable G"odel algebras.

A (quasi-)equational definition of the rotated structures is provided.

Abstract

The present paper is devoted to study the effect of connected and disconnected rotations of G\"odel 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 G\"odel algebras endowed with modal operators.