S-preclones and the Galois connection ^SPol–^SInv, Part I

Authors

Peter Jipsen, Chapman UniversityFollow
Erkko Lehtonen, Khalifa University of Science and Technology
Reinhard Pöschel, TU Dresden

Document Type

Article

Publication Date

7-9-2024

Abstract

We consider S-operations f : Aⁿ → A in which each argument is assigned a signum 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 = (s)s∈S, S-relational clones, and a preservation property (f S), and we consider the induced Galois connection ^SPol–^SInv. 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.

Comments

This article was originally published in Algebra Universalis, volume 85, in 2024. https://doi.org/10.1007/s00012-024-00863-7

Recommended Citation

Jipsen, P., Lehtonen, E. & Pöschel, R. S-preclones and the Galois connection ^SPol–^SInv, Part I. Algebra Univers. 85, 34 (2024). https://doi.org/10.1007/s00012-024-00863-7

Peer Reviewed

1

Copyright

The authors

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.