Picture Fuzzy Subalgebras and Ideals in Sheffer Stroke UP-Algebras
Manish Kumar Gunjan *
Department of Mathematics, Lalit Narayan Mithila University, India.
Amal Kumar Adak
Department of Mathematics, Ganesh Dutt College, Begusarai, India.
Wajid Ali
Department of Mathematics, Air University, Islamabad, Pakisthan.
*Author to whom correspondence should be addressed.
Abstract
Picture fuzzy sets, introduced by Cuong in 2014, extend intuitionistic fuzzy sets by assigning three independent degrees to each element-positive membership (μ), neutral membership (η), and negative membership (ν), together with an explicit refusal degree, thereby enabling a richer representation of uncertainty involving neutrality and abstention. In this paper, picture fuzzy structures are introduced into Sheffer stroke UP-algebras (SUP-algebras), an algebraic framework based on the functionally complete Sheffer stroke (NAND) operation. Picture fuzzy SUP-subalgebras and picture fuzzy SUP-ideals are defined, and several equivalent characterizations are established using level sets, complement functions, and characteristic picture fuzzy sets associated with crisp subalgebras and ideals. It is shown that every picture fuzzy SUP-ideal is a picture fuzzy SUP-subalgebra, while the converse does not hold in general, and that the family of all picture fuzzy SUP-subalgebras forms a complete distributive lattice. Preservation properties under direct products and surjective homomorphisms are also investigated. These results enrich the theory of SUP-algebras by incorporating neutrality and
refusal into their algebraic structure, thereby enhancing their applicability to uncertainty modeling, non-classical logical systems, and decision-making frameworks.
Keywords: Sheffer stroke UP-algebra, picture fuzzy set, picture fuzzy SUP-subalgebra, picture fuzzy SUP-ideal, level set, direct product, homomorphism