Navigating Bipolar Indeterminacy: Bipolar IndetermSoft Sets and
Bipolar IndetermHyperSoft Sets for Knowledge Representation
Takaaki Fujita1,∗ , Florentin Smarandache2
1Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan
2University of New Mexico, Gallup Campus, NM 87301, USA
Emails: Takaaki.fujita060@gmail.com; smarand@unm.edu
Abstract
A variety of mathematical frameworks—such as fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, soft
sets, rough sets, and plithogenic sets—have been developed to model uncertainty, with wide applications in
decision making, data analysis, and artificial intelligence. Within soft set theory, extensions like hypersoft
sets, indeterm-soft sets, indeterm-hypersoft sets, bipolar soft sets, and bipolar hypersoft sets have further
enhanced its expressive power. In this paper, we introduce two new constructs: bipolar indeterm-soft sets and
bipolar indeterm-hypersoft sets. We provide their formal definitions, establish key algebraic properties, and
demonstrate how they naturally combine bipolar evaluation with inherent indeterminacy. These models offer
a versatile toolkit for capturing complex forms of uncertainty and lay the groundwork for future theoretical
advances and practical applications in soft set theory.
Keywords: Soft Set; Indeterm-Soft Set; Hypersoft set; Indeterm-HyperSoft Set; Bipolar Soft Set; Bipolar
Hypersoft Set