Algebraic structures of circular intuitionistic fuzzy triples: Monoids, multi-unitary groups, and lattice properties

This paper investigates the algebraic structures induced by binary operations on Circular Intuitionistic Fuzzy Triples (C-IFTs). For a fixed upper bound on the circular radius, the neutral and absorbing elements of the standard union, intersection, algebraic addition, and algebraic multiplication are determined, considering both minimum and maximum aggregation of the radius components. It is proved that the corresponding algebraic structures are commutative monoids and commutative multi-unitary groups (μ-groups), although they do not form ordinary groups on the entire domain of C-IFTs. The sets of μ-inverse elements are described explicitly, and the dual μ-groups are determined. Furthermore, it is established that the standard union with maximum radius aggregation and intersection with minimum radius aggregation form a bounded distributive lattice. The relationships between the operations are examined under the standard radius-preserving negation. It is shown that the duality generated by the negation preserves the radius operation, whereas the duality of μ-groups requires the opposite one. The results characterize the algebraic properties of the selected C-IFT operations and provide a basis for further study of aggregation procedures and circular intuitionistic fuzzy index-matrix constructions.