Polifunctions: A Formal Framework for Multi-Valued Mathematical Functions

This paper introduces a novel formal framework for functions that can simultaneously return multiple values, sets, intervals, or other complex mathematicalstructures for a single input. We define the concept of a ”polifunction” as a generalization of traditional functions, multi-valued functions, and relations. This framework bridges the gap between mathematical theory and practical applications incomputer science and physics, where returning multiple values is often natural ornecessary. We develop a comprehensive algebra for operations on polifunctions andexplore their properties. The proposed framework has potential applications invarious fields, including quantum mechanics, complex analysis, and computationalmathematics, where traditional single-valued functions may be insufficient to modelintrinsic multiplicity or simultaneous outcomes.