On Graph Products of Automatic and Biautomatic Monoids

It is shown that the graph product of automatic monoids is always automatic thereby improving on a result by Veloso da Costa 22] who showed this result provided the factors have finite geometric type. Secondly, we prove that, in general, the free product (and therefore the graph product) of biautomatic monoids need not be biautomatic. Imposing a restriction on the factors that is symmetric to Veloso da Costa's "finite geometric type", the biautomaticity of all graph products of biautomatic monoids is shown.