Plithogenic Directed Offset and Plithogenic MultiDirected Offsets

Abstract. For more than half a century, concepts such as fuzzy sets have been studied to handle uncertainty
in real-world systems. Uncertainty-modeling frameworks—such as fuzzy sets, intuitionistic fuzzy sets, hyperfuzzy
sets, neutrosophic sets, hyperneutrosophic sets, soft sets, rough sets, and plithogenic sets—have been
widely adopted to represent and reason under vagueness and imprecision. Plithogenic sets extend these approaches
by combining multi-dimensional degrees of membership with a measure of contradiction. The offset
enhancement further allows membership values to lie outside the usual [0, 1] interval, accommodating over- or
under-membership. In this paper, we introduce and rigorously define the Plithogenic Multi-Directed Offset, unifying
the concepts of uncertainty, directionality, and non-standard memberships; we show that this framework
subsumes existing plithogenic, directed, and offset structures.

Keywords: MultiDirected set, Fuzzy Set, Directed Set, Plithogenic Offset, Plithogenic Set