Locally Integral Involutive PO-Semigroups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915698789318656 |
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| author | Gil-Férez, José Jipsen, Peter Sugimoto, Melissa |
| author_facet | Gil-Férez, José Jipsen, Peter Sugimoto, Melissa |
| contents | We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $Φ= \{φ_{pq}: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_Φ\mathbf A_p$ of its integral components along $Φ$. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $Φ$ is an ipo-semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12926 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Locally Integral Involutive PO-Semigroups Gil-Férez, José Jipsen, Peter Sugimoto, Melissa Logic 06F05 (Primary) 03G10, 06B15, 06E75 (Secondary) We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $Φ= \{φ_{pq}: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_Φ\mathbf A_p$ of its integral components along $Φ$. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $Φ$ is an ipo-semigroup. |
| title | Locally Integral Involutive PO-Semigroups |
| topic | Logic 06F05 (Primary) 03G10, 06B15, 06E75 (Secondary) |
| url | https://arxiv.org/abs/2310.12926 |