The criteria for the uniqueness of a weight homomorphism of a baric algebra

TL;DR

This paper establishes criteria for the uniqueness of weight homomorphisms in baric algebras, including conditions involving systems of equations and transition matrices, and provides counterexamples to existing conditions.

Contribution

It introduces new necessary and sufficient criteria for the uniqueness of weight homomorphisms in baric algebras, expanding understanding beyond previous conditions.

Findings

A system of equations criterion for uniqueness

Counterexample to Holgate's sufficient condition

Transition matrix criterion for uniqueness

Abstract

The criteria for a baric algebra $A$ (over a field $K$) to have a unique weight homomorphism are found. One of them requires a certain system of equations to have a unique non-trivial solution in the field $K$. Applying this criterion, we provide an example showing that Holgate's well-known sufficient condition for the uniqueness of a weight homomorphism is not necessary, and give also a new example of a baric algebra with two weight homomorphisms. Another criterion found in this paper asserts that a baric algebra has a unique weight homomorphism if and only if the transition matrix from any semi-natural basis $B_1$ to any semi-natural basis $B_2$ is stochastic.