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.

翻译：本文致力于研究基于直接不可分解结构的、带有算子的Gödel代数在连通旋转与非连通旋转下的效应。通过此构造所呈现的结构是幂零极小代数（根据旋转是连通或非连通，可能包含或不包含否定不动点），其上的特殊模态算子定义在一个直接不可分解代数上。本文将给出这些后述结构的（拟）等式定义。我们的主要结果表明，带有模态算子的直接不可分解幂零极小代数（无论是否包含否定不动点）可被完全刻画为带有模态算子的直接不可分解Gödel代数的连通与非连通旋转。