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

• Published in: Algebra universalis Vol. 85; no. 3
• Main Authors: Jipsen, Peter, Lehtonen, Erkko, Pöschel, Reinhard
• Format: Journal Article
• Language: English
• Published: 01.08.2024
• ISSN: 0002-5240; 1420-8911
• DOI: 10.1007/s00012-024-00863-7

Abstract We consider S - operations $$f :A^{n} \rightarrow A$$ f : A n → A in which each argument is assigned a signum $$s \in S$$ s ∈ 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 $$\varrho = (\varrho _{s})_{s \in S}$$ ϱ = ( ϱ s ) s ∈ S , S - relational clones , and a preservation property ("Equation missing" ), and we consider the induced Galois connection $^{S}{}\textrm{Pol}$$ S Pol – $^{S}{}\textrm{Inv}$$ S 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 .