$2\times 2$ monotone grid classes are finitely based

TL;DR

This paper proves that all 2x2 monotone grid classes can be characterized by a finite set of minimal forbidden permutations, extending to certain generalized grid classes with specific monotone cell arrangements.

Contribution

It establishes the finite basis property for all 2x2 monotone grid classes, including a broader class with two monotone cells in the same row.

Findings

All 2x2 monotone grid classes are finitely based.

Generalized grid classes with two monotone cells in the same row are also finitely based.

Abstract

In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.