# On discrete idempotent paths

## TL;DR

This paper characterizes idempotent paths on a lattice as upper zigzag paths, linking their enumeration to odd Fibonacci numbers, and provides a geometric proof for counting idempotents in certain monoids.

## Contribution

It explicitly describes the monoid structure of discrete paths and characterizes idempotent paths using join-continuous maps, connecting combinatorics with algebraic structures.

## Key findings

- Idempotent paths are upper zigzag paths.
- Number of idempotent paths corresponds to odd Fibonacci numbers.
- Provides a geometric proof for counting idempotents in monoids.

## Abstract

The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain { 0, 1,. .. , n } to itself. We explicitly describe this monoid structure and, relying on a general characterization of idempotent join-continuous maps from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomaps on finite chains.