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?



Product of two algebras with maximum condition
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.


