The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory - Krystian Jobczyk (IEEE Access, 2022)
A category theory constitutes a convenient conceptual apparatus to organize the worlds of mathematical entities. The concept of fuzzy natural transformation as an abstract mapping on functors is one of the essential concepts of this theory. If we admit a piece of non-commutativity in its definition diagrams, then the ‘upward’ and the ‘downward’ diagram parts generate different result sets. In this way, we can introduce the concept of fuzzy natural transformation. We deal with the multi-fuzzy natural transformation if such a transformation is based on a multi-diagram. This paper aims to describe the multi-fuzzy natural transformation situation when the symmetric difference of the result is finite. It allows us to organize the whole spectrum of the result sets in a unique quotient algebra $\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ . Different algebraic properties of this structure will be explored, and a piece of classical encoding theory will be reconstructed in the environment determined by this algebra. In particular, the concepts of the $k$ -multi similarity, the abstract $k$ -multi similarity, the $k$ -similarity balls, and the abstract $k$ -similarity balls are introduced as some generalization of the idea of Hamming’s distances and Hamming’s balls. Finally, it is shown how to automate some verification processes in this context using an R-based programming environment.