Structure of Chinese algebras

TL;DR

This paper investigates the algebraic structure of the Chinese monoid's algebra over a field, describing its prime ideals, radicals, and embeddings, and providing a new representation that shows the monoid satisfies a nontrivial identity.

Contribution

It characterizes the minimal prime ideals, establishes the equality of the prime radical and Jacobson radical, and introduces a novel embedding of the monoid into a product of simpler monoids.

Findings

Minimal prime ideals correspond to certain homogeneous congruences.

Prime radical equals the Jacobson radical in the algebra.

Monoid embeds into a product of bicyclic and infinite cyclic monoids.

Abstract

The structure of the algebra K[M] of the Chinese monoid M over a field K is studied. The minimal prime ideals are described. They are determined by certain homogeneous congruences on M and they are in a one to one correspondence with diagrams of certain special type. There are finitely many such ideals. It is also shown that the prime radical B(K[M]) of K[M] coincides with the Jacobson radical and the monoid M embeds into the algebra K[M]/B(K[M]). A new representation of M as a submonoid of the direct product of finitely many copies of the bicyclic monoid and finitely many copies of the infinite cyclic monoid is derived. Consequently, M satisfies a nontrivial identity.