On the lattice of the weak factorization systems on a finite lattice

TL;DR

This paper studies the structure of weak factorization systems on finite lattices, proving their semidistributivity, trimness, and congruence uniformity, and applies these findings to enumerate transfer systems.

Contribution

It establishes fundamental lattice-theoretic properties of weak factorization systems and introduces a graph-based method for counting transfer systems on finite lattices.

Findings

The lattice of weak factorization systems is semidistributive, trim, and congruence uniform.

A graph theoretical approach to enumerate transfer systems is developed.

A lower bound for the number of transfer systems on a boolean lattice is provided.

Abstract

We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer systems. As an application we find a lower bound for the number of transfer systems on a boolean lattice.