Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912150138650624 |
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| author | Mio, Matteo Sarkis, Ralph Vignudelli, Valeria |
| author_facet | Mio, Matteo Sarkis, Ralph Vignudelli, Valeria |
| contents | We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14361 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces Mio, Matteo Sarkis, Ralph Vignudelli, Valeria Logic in Computer Science We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation. |
| title | Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2304.14361 |