Which properties are inherited by the Cartesian product of two sets equipped with a binary operation?

Question

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?

to better frame the question, can you point to a property that $G$ and $H$ have, but $G\times H$ does not have? — Ittay Weiss, Mar 14 '13 at 09:04
@IttayWeiss If $G$ and $H$ are simple groups $G\times H$ isn't. (I don't know whether simplicity is a "thing" in magmas in general, but groups are magmas, so this is one example.) — Alexander Gruber, Mar 14 '13 at 09:11
I'm not sure that is what OP had in mind, hence my question. — Ittay Weiss, Mar 14 '13 at 09:12
@IttayWeiss Sure. Consider the multiplicative structure of $\mathbb{R}$. This fails to form a group, because PRECISELY ONE element of $\mathbb{R}$ fails to have a multiplicative inverse. Now consider the magma $\mathbb{R} \times \mathbb{R}$. It possesses MANY elements that fail to have inverses. e.g. all the elements of the form $(x,0)$ and $(0,y)$. So the property of having a unique non-invertible element is not preserved. — goblin GONE, Mar 14 '13 at 09:12
If the property is expressed by a formula $f(a,b,c,...)$ with $a,b,c$ being arbitrary unrelated elements of the group, and it doesn't matter which group ($G$ or $H$) you choose, then it will hold for $G\times H$ as well. E.g. $a(bc)=(ab)c$ might hold for both. While $a^3=e$, $e$ being the respective group unit, or $a^2=a^{-1}$, might not hold for both. Then $(.,.)^3=(e_G,e_H)$ doesn't hold. — Nikolaj-K, Mar 14 '13 at 09:13

score 5 · Accepted Answer · answered Mar 14 '13 at 09:43

5

By Birkhof Theorem every variety of (universal) algebras is closed respectively direct product. So if an identity is true for $A$ and $B$, it is true for $A\times B$.

answered Mar 14 '13 at 09:43

Boris Novikov

17,754

An identity being a statement like $\forall a,b \in A(ab=ba)$? Are we allowed to use "OR" and "There exists"? – goblin GONE Mar 14 '13 at 10:48
No, only $\forall$. – Boris Novikov Mar 14 '13 at 11:28
Conversely, the direct factors, being homomorphic images of the product, inherit all equational properties of the product. – James Mar 16 '13 at 02:25
@James: Certainly, but I only answered thr question. – Boris Novikov Mar 16 '13 at 06:49
@Boris: The question also says "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?" (I haven't written anything about that in my answer yet, either.) – Tara B Mar 18 '13 at 13:08
@Tara B: My answer means: all properties which follow from identities, are inherited (in both directions). – Boris Novikov Mar 18 '13 at 13:43
I agree that that is true, but while your answer may mean it, it only actually mentions one direction. – Tara B Mar 18 '13 at 18:56
@Tara B: If an identity is true for $G\times H$ then it is true for $G$, isn't it? – Boris Novikov Mar 18 '13 at 19:55
Yes. I already said I agreed with that, but you haven't mentioned it in your answer. – Tara B Mar 18 '13 at 22:01
@ Tara B: Sorry! – Boris Novikov Mar 18 '13 at 22:05

Tara B · Answer 2 · 2013-03-14T13:06:47.797

I am quite certain that there is no currently known general rule which dictates exactly which properties $G\times H$ will inherit from $G$ and $H$, nor one which dictates which properties $G$ and $H$ inherit from $G\times H$. If there were, I would not have seen so many results on special cases in recent publications (not proved by referring to some general principle).

Here are some examples of properties not (necessarily) inherited by $G\times H$:

Freeness
Finite generation: This is inherited in magmas with identity, but not always otherwise. If $S$ and $T$ are f.g. semigroups, then $S\times T$ is f.g. iff either at least one of $S$ and $T$ is finite or $S^2=S$ and $T^2=T$. (This result is due to Ruskuc, Robertson and Wiegold.)
Having word problem in a specific complexity class. For example the infinite cyclic group $\mathbb{Z}$ has context-free word problem, but $\mathbb{Z}\times \mathbb{Z}$ does not. Similarly, the semigroup $\mathbb{N}$ of natural numbers under addition has rational word problem, but $\mathbb{N}\times \mathbb{N}$ does not.

I'll post this for now, but might come back and add to it later, since I haven't by any means addressed everything in your question.

@user18921: I like your question and could say more about it, but it is quite broad. Could you please let me know in which direction it would be most helpful to extend my answer? — Tara B, Mar 15 '13 at 23:30

score 2 · Answer 3 · edited Mar 14 '13 at 09:09

2

Some points I know:

If $G$ and $H$ are semigroups, so is $G\times H$ since:

$$[(a,b)(c,d)](h,f)=(ac,bd)(h,f)=[(ac)h,(bd)f]=[a(ch),b(df)]=...(a,b)[(c,d),(h,f)]$$

An element $(a,b)\in G\times H$ is idempotent iff $a$ is an idempotent in $G$ and $b$ is an idempotent in $H$.
An element $(a,b)\in G\times H$ is a left (or right) identity element of $G\times H$ iff $a$ is a left (or right) identity element of $G$ and $b$ is a left (or right) identity element of $H$

edited Mar 14 '13 at 09:09

Alexander Gruber

28,037

answered Mar 14 '13 at 09:06

Mikasa

67,942

Helpful reminder of key definitions! +1 – amWhy Mar 15 '13 at 01:32
@amWhy: Thanks a lot. :-) – Mikasa Mar 15 '13 at 05:26

Which properties are inherited by the Cartesian product of two sets equipped with a binary operation?

3 Answers3