# 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.

## Key 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.