Lift and generalized ordinal sum of negations on bounded posets

Abstract

In this paper, two construction methods for fuzzy negations on bounded structures are presented. The first method focuses on lifting a fuzzy negation from a more general subposet to the complete lattice, extending the existing research. It provides an effective approach to lift fuzzy negations, even when the top and bottom elements of the subposet differ from those of the complete lattice. The second method explores the generalized ordinal sum of negations on an abelian partially ordered group, achieved by complementing a given fuzzy negation. A sufficient and necessary condition for the generalized ordinal sum, with a familiar fuzzy negation as the complement, to be a fuzzy negation is given. Building upon this, the paper introduces a supplementary fuzzy negation based on the provided family of subintervals, which guarantees that the generalized ordinal sum is still a fuzzy negation.