Associative, idempotent, symmetric, and order-preserving operations on chains

TL;DR

This paper characterizes specific algebraic operations on finite chains, revealing their structure and counting them using Catalan numbers, thus advancing the understanding of order-preserving operations.

Contribution

It provides a complete characterization of associative, idempotent, symmetric, and order-preserving operations on finite chains and links their count to Catalan numbers.

Findings

Number of such operations on an n-element chain equals the nth Catalan number.

Characterization of these operations via properties of their associated semilattice order.

Insight into the structure of order-preserving operations on chains.

Abstract

We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an $n$-element chain is the $n^{\text{th}}$ Catalan number.