Suppose $A$ and $B$ are two algebras of the same signature, both having maximum condition on sub-algebras. Is it true that $A\times B$ has the same property?
Here is a semigroup example.
Take the semigroup $\mathbb Z_+$ of positive integers. Each of its subsemigroups is finitely generated so it has the maximum condition on subsemigroups. But $\mathbb Z_+\times \mathbb Z_+$ is not finitely generated so it does not have the maximum condition.
Indeed, any element of the form $(m,1)$ with $m>0$ is not a sum of two elements of $\mathbb Z_+\times \mathbb Z_+$. Thus these elements must belong to any generating set.