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arXiv · 2306.00493

$S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I

Peter Jipsen·Erkko Lehtonen·Reinhard Pöschel
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Abstract

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 $A$.

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Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel. 2023-06-01. $S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I. https://doi.org/10.1007/s00012-024-00863-7

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