Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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
1  
What is maximum condition? –  The Masked Avenger Feb 19 at 22:38
2  
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 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.

share|improve this answer
    
Thank you, it seems that determining the type of algebras having this property must be hard. –  M. Shahryari Feb 20 at 3:06

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.