$S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916498833932288 |
|---|---|
| author | Jipsen, Peter Lehtonen, Erkko Pöschel, Reinhard |
| author_facet | Jipsen, Peter Lehtonen, Erkko Pöschel, Reinhard |
| contents | We consider $S$-operations $f \colon A^{n} \to A$ in which each argument is assigned a signum $s \in S$ representing a "property" such as being order-preserving or order-reversing with respect to a fixed partial order on $A$. The set $S$ of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of $S$-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all $S$-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of $S$-preclone. We introduce $S$-relations $\varrho = (\varrho_{s})_{s \in S}$, $S$-relational clones, and a preservation property ($f \mathrel{\stackrel{S}{\triangleright}} \varrho$), and we consider the induced Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$. The $S$-preclones and $S$-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all $S$-preclones on $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00493 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I Jipsen, Peter Lehtonen, Erkko Pöschel, Reinhard Rings and Algebras We consider $S$-operations $f \colon A^{n} \to A$ in which each argument is assigned a signum $s \in S$ representing a "property" such as being order-preserving or order-reversing with respect to a fixed partial order on $A$. The set $S$ of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of $S$-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all $S$-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of $S$-preclone. We introduce $S$-relations $\varrho = (\varrho_{s})_{s \in S}$, $S$-relational clones, and a preservation property ($f \mathrel{\stackrel{S}{\triangleright}} \varrho$), and we consider the induced Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$. The $S$-preclones and $S$-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all $S$-preclones on $A$. |
| title | $S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2306.00493 |