# The distinguishing number and distinguishing chromatic number for posets

## TL;DR

This paper introduces the concepts of the distinguishing number and chromatic number for posets, providing bounds and specific results for certain classes of lattices, advancing understanding of symmetry-breaking in poset structures.

## Contribution

It defines new invariants for posets and establishes bounds, including for Boolean and divisibility lattices, using lattice theory techniques.

## Key findings

- Any linear extension of the set of join-irreducibles generates a 2-coloring.
- Upper bounds for the distinguishing chromatic number are established.
- The distinguishing number of twin-free Cohen-Macaulay planar lattices is at most 2.

## Abstract

In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory to show that any linear extension of $Q_L$ generates a distinguishing 2-coloring of $L$. We prove general upper bounds for the distinguishing chromatic number and particular upper bounds for the Boolean lattice and for divisibility lattices. In addition, we show that the distinguishing number of any twin-free Cohen-Macaulay planar lattice is at most 2.