The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory
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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
IEEE
2022-01-01
|
| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/9695485/ |
| _version_ | 1811342201693667328 |
|---|---|
| author | Krystian Jobczyk |
| author_facet | Krystian Jobczyk |
| author_sort | Krystian Jobczyk |
| collection | DOAJ |
| description | 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-similarity balls, and the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-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. |
| first_indexed | 2024-04-13T19:07:21Z |
| format | Article |
| id | doaj.art-c061f6240d6c45cc9028944b31a47a61 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-04-13T19:07:21Z |
| publishDate | 2022-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-c061f6240d6c45cc9028944b31a47a612022-12-22T02:33:57ZengIEEEIEEE Access2169-35362022-01-0110151351515010.1109/ACCESS.2022.31471659695485The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding TheoryKrystian Jobczyk0https://orcid.org/0000-0001-6194-2737Department of Applied Computer Science, AGH University of Science and Technology, Kraków, PolandA 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-similarity balls, and the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-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.https://ieeexplore.ieee.org/document/9695485/The fuzzy natural transformationP(ω)/fingeneralized encoding theoryHamming distancesR programming |
| spellingShingle | Krystian Jobczyk The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory IEEE Access The fuzzy natural transformation P(ω)/fin generalized encoding theory Hamming distances R programming |
| title | The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory |
| title_full | The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory |
| title_fullStr | The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory |
| title_full_unstemmed | The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory |
| title_short | The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory |
| title_sort | fuzzy natural transformations the algebra p x03c9 fin and generalized encoding theory |
| topic | The fuzzy natural transformation P(ω)/fin generalized encoding theory Hamming distances R programming |
| url | https://ieeexplore.ieee.org/document/9695485/ |
| work_keys_str_mv | AT krystianjobczyk thefuzzynaturaltransformationsthealgebrapx03c9finandgeneralizedencodingtheory AT krystianjobczyk fuzzynaturaltransformationsthealgebrapx03c9finandgeneralizedencodingtheory |