Take the 2-minute tour ×  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?
share|improve this question What is maximum condition? –  The
Masked Avenger Feb 19 at 22:38 Take ${\mathbb Z}$ with two unary
operations $i:z\mapsto z+1$ and $d:z\mapsto z-1$. Such an
algebra has only two subalgebras $\varnothing$ and ${\mathbb
Z}$. Whereas in ${\mathbb Z}\times{\mathbb Z}$, a subalgebra is
any subset closed with respect to the action of the group
$({\mathbb Z},+)$ given by $z\cdot(z_1,z_2):=(z+z_1,z+z_2)$. Any
orbit is given by the number $z_1-z_2$. Therefore, a subalgebra
is nothing but a collection of integer numbers. Consequently,
the maximality condition does not hold. –  Sasha Anan'in Feb 19
at 22:57  1 Answer 1  up vote 1 down vote accepted  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