Finitely generated maximal partial clones and their intersections

TL;DR

This paper investigates the finite generation properties of certain partial clones on finite sets, revealing differences between small and larger sets and the complexity of their intersections.

Contribution

It demonstrates that on a two-element set, certain partial clones are not finitely generated but are intersections of finitely generated maximal partial clones, and for larger sets, intersections can be non-finitely generated.

Findings

For |A|=2, the selfdual monotone partial functions form a non-finitely generated partial clone.

On larger sets, intersections of finitely generated maximal partial clones can be non-finitely generated.

The structure of partial clones varies significantly with the size of the underlying set.

Abstract

Let A be a finite non-singleton set. For |A|=2 we show that the partial clone consisting of all selfdual monotone partial functions on A is not finitely generated, while it is the intersection of two finitely generated maximal partial clones on A. Moreover for |A| >= 3 we show that there are pairs of finitely generated maximal partial clones whose intersection is a non-finitely generated partial clone on A.