The algebra of the Catalan monoid as an incidence algebra: A simple proof

TL;DR

This paper provides a simple, direct proof establishing an isomorphism between the algebra of the Catalan monoid and an incidence algebra of a specific poset, enhancing understanding of their algebraic structure.

Contribution

It introduces a straightforward proof of the algebraic isomorphism between the Catalan monoid algebra and an incidence algebra, clarifying their relationship.

Findings

Proves the isomorphism between Catalan monoid algebra and incidence algebra

Simplifies understanding of the algebraic structure of the Catalan monoid

Applicable over any commutative ring with identity

Abstract

We give a direct straightforward proof that there is an isomorphism between the algebra of the Catalan monoid C_n that is, the monoid of all order-preserving, weakly increasing self-maps f of [n] = {1,...,n}, over any commutative ring with identity and the incidence algebra of a certain poset over the ring.