Let $G$ and $H$ denote sets equipped with a binary operation (aka magmas). We can form the Cartesian product magma $G \times H$ in the obvious way. I'm interested in which properties of $G$ and $H$ transfer to $G \times H$. For instance, if both $G$ and $H$ are associative, then $G \times H$ is associative. Similarly with commutativity.
Is there a general principle that dictates which properties $G \times H$ will inherit?
And what about the other way around? For which properties does it hold that if $G \times H$ has that property, then either/both of $G, H$ must have it?