On complex fuzzy multisets

Abstract

This article proposes a mathematical structure called complex fuzzy multisets (CFMSs), which integrates the multiplicity feature of fuzzy multisets with the complex-valued membership grades of complex fuzzy sets. In this framework, each membership grade is expressed as re^iφ, with r ∈ [0, 1] and φ ∈ [0, 2π), enabling simultaneous representation of membership intensity, phase information, and multiple occurrences of elements. We provide a formal definition of CFMSs and establish fundamental operations, including inclusion, complement, intersection, union, Cartesian product, rotation, and reflection. The algebraic analysis proves that CFMSs satisfy involution, idempotence, commutativity, associativity, distributivity, De Morgan's laws, and absorption, thereby forming a distributive lattice with an involution. Additionally, we define the image and inverse image of CFMSs under mappings. This work lays a rigorous foundation for further theoretical development and potential applications in multi-source data fusion, periodic signal processing, and reasoning under uncertainty with repeated observations.