Shadowed Offset: Integrating Offset and Shadowed Set Frameworks for Enhanced Uncertainty Modeling

Abstract

Uncertainty‑handling frameworks—including Fuzzy Sets, Intuitionistic Fuzzy Sets, Hyperfuzzy Sets, Neutrosophic Sets, HyperNeutrosophic Sets, Soft Sets, Rough Sets, and Plithogenic Sets—have become fundamental tools for modeling imprecision and vagueness across diverse application domains. Classical fuzzy sets extend traditional set theory by assigning each element a membership degree within the unit interval [0,1], thereby capturing graded inclusion. To represent under‑ and over‑membership, many of these theories have been enriched with offset concepts that allow membership values to fall outside this canonical range. Shadowed sets complement these developments by employing two thresholds to partition the membership spectrum into three regions—excluded, included, and indeterminate—providing a compact mechanism for residual uncertainty. Despite the complementary strengths of offset and shadowed approaches, a unified formalism integrating both has not yet been explored. In this paper, we introduce the novel concept of the Shadowed Offset, which extends the shadowed‑set paradigm by permitting off‑interval membership while preserving tripartite thresholding. We present a rigorous mathematical definition of Shadowed Offsets, derive their fundamental properties, and illustrate their behavior through representative examples. The proposed construct offers a more flexible and expressive framework for complex uncertainty, and we conclude by outlining promising directions for future research.