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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2410.00604 |
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| _version_ | 1866912053413806080 |
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| author | Bonzio, Stefano Gil-Férez, José Jipsen, Peter Přenosil, Adam Sugimoto, Melissa |
| author_facet | Bonzio, Stefano Gil-Férez, José Jipsen, Peter Přenosil, Adam Sugimoto, Melissa |
| contents | A residuated poset is a structure $\langle A,\le,\cdot,\backslash,/,1 \rangle$ where $\langle A,\le \rangle$ is a poset and $\langle A,\cdot,1 \rangle$ is a monoid such that the residuation law $x\cdot y\le z\iff x\le z/y\iff y\le x\backslash z$ holds. A residuated poset is balanced if it satisfies the identity $x\backslash x \approx x/x$. By generalizing the well-known construction of Plonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two families of maps (instead of one, as in the usual case), we construct a residuated poset based on the disjoint union of their domains. We apply this approach to provide a structural description of some varieties of residuated lattices and relation algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of balanced residuated partially ordered monoids Bonzio, Stefano Gil-Férez, José Jipsen, Peter Přenosil, Adam Sugimoto, Melissa Logic A residuated poset is a structure $\langle A,\le,\cdot,\backslash,/,1 \rangle$ where $\langle A,\le \rangle$ is a poset and $\langle A,\cdot,1 \rangle$ is a monoid such that the residuation law $x\cdot y\le z\iff x\le z/y\iff y\le x\backslash z$ holds. A residuated poset is balanced if it satisfies the identity $x\backslash x \approx x/x$. By generalizing the well-known construction of Plonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two families of maps (instead of one, as in the usual case), we construct a residuated poset based on the disjoint union of their domains. We apply this approach to provide a structural description of some varieties of residuated lattices and relation algebras. |
| title | On the structure of balanced residuated partially ordered monoids |
| topic | Logic |
| url | https://arxiv.org/abs/2410.00604 |