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

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.