# Smooth digraphs modulo primitive positive constructability and cyclic loop conditions

## TL;DR

This paper characterizes the structure of finite smooth digraphs under pp-constructability, showing the poset forms a distributive lattice and linking separability to prime cyclic loop conditions in polymorphism clones.

## Contribution

It provides a complete description of the pp-constructability poset for smooth digraphs and relates it to cyclic loop conditions in polymorphism clones.

## Key findings

- The pp-constructability poset of smooth digraphs is a distributive lattice.
- Separation of digraphs in the poset can be achieved via prime cyclic loop conditions.
- The poset of cyclic loops ordered by strength also forms a distributive lattice.

## Abstract

Finite smooth digraphs, that is, finite directed graphs without sources and sinks, can be partially ordered via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two smooth digraphs in our poset it suffices to show that the polymorphism clone of one of the digraphs satisfies a prime cyclic loop condition that is not satisfied by the polymorphism clone of the other. Furthermore, we prove that the poset of cyclic loop ordered by their strength for clones is a distributive lattice, too.