# Congruences on Orthogonal Rook Monoids and Symplectic Rook Monoids

## TL;DR

This paper classifies all nonuniform congruences on orthogonal and symplectic rook monoids, revealing multiple types depending on the size and structure of the monoids, with explicit descriptions in algebraic terms.

## Contribution

It provides a complete classification of nonuniform congruences on orthogonal and symplectic rook monoids, including explicit descriptions and special cases for small n.

## Key findings

- Four types of nonuniform congruences on OR_n for even n≠4.
- Six types of nonuniform congruences on OR_4.
- Only one type of congruence on symplectic rook monoids for all even n≥2.

## Abstract

We give a complete classification of all nonuniform congruences on orthogonal rook monoids and symplectic rook monoids. We find that there are four kinds of nonuniform congruences on the orthogonal rook monoids ${OR}_n$ for even $n\ne 4$, and we describe each kind of the congruences explicitly in terms of normal subgroups of maximal subgroups. We also find that if $n = 4$, there are six kinds of nonuniform congruences on ${OR}_4$, and we describe these congruences using both $\mathcal{H}$-relations and certain normal subgroups of some maximal subgroups. In contrast, we find that there is only one kind of congruences on the symplectic rook monoids for all even $n\ge 2.$