S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{S}$$\end{document}-preclones and the Galois connection SPol\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{}^{S}{}\textrm{Pol}}$$\end{document}–SInv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{}^{S}{}\textrm{Inv}}$$\end{document}, Part I

We consider S-operationsf:An→A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :A^{n} \rightarrow A$$\end{document} in which each argument is assigned a signums∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in S$$\end{document} 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-relationsϱ=(ϱs)s∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho = (\varrho _{s})_{s \in S}$$\end{document}, S-relational clones, and a preservation property ([inline-graphic not available: see fulltext]), and we consider the induced Galois connection SPol\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{S}{}\textrm{Pol}$$\end{document}–SInv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{S}{}\textrm{Inv}$$\end{document}. 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.