# Large filters of quasiorder lattices can be generated by few elements

## TL;DR

This paper proves that for small posets, the filter of quasiorders extending the poset order can be generated by a few elements, revealing a structural simplicity in such lattices.

## Contribution

It establishes that quasiorder filters of small posets are finitely generated, providing new insights into the structure of these lattices.

## Key findings

- Filters can be generated by few elements for small posets
- Structural simplicity of quasiorder lattices in small posets
- Advances understanding of lattice generation in order theory

## Abstract

For a poset $(P;\leq)$, the quasiorders (AKA preorders) extending the poset order "$\leq$" form a complete lattice $F$, which is a filter in the lattice of all quasiorders of the set $P$. We prove that if the poset order "$\leq$" is small, then $F$ can be generated by few elements.