Title: $S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I Show Chinese title

Title: $S$-前克隆和伽罗瓦连接${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$，第一部分

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 $A$. Show Chinese abstract

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). 所有具有特定性质的$S$操作，对于其带符号的参数而言，不是一个克隆（因为其不封闭于任意参数的识别），但它是一个具有特殊性质的前克隆，这导致了$S$前克隆的概念。 我们引入$S$-关系$\varrho = (\varrho_{s})_{s \in S}$，$S$-关系克隆，以及一个保持性质 ($f \mathrel{\stackrel{S}{\triangleright}} \varrho$)，并考虑由此产生的伽罗瓦连接${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$。 $S$-前克隆和$S$-关系克隆恰好是这个伽罗瓦连接的闭集。 我们还建立了一些关于所有$S$-前克隆在$A$上的格结构的基本事实。