GCD on a union magma

1. Feb 22, 2014

1MileCrash

1. The problem statement, all variables and given/known data

Consider the following magma, S is not empty; P(S) is the power set.

(P(S), U)

Now, let A and B be in P(S).

What is the GCD of A and B?

2. Relevant equations

3. The attempt at a solution

If I choose a common divisor of A and B under unions, call it X, I get that X is a subset of A and B as the only requirement.

The only way I can interpret "GCD" is that X is the largest such set. Then X = (A intersect B.)

Agree?

2. Feb 22, 2014

Dick

You should give a few more definitions here. But it sounds plausible if 'divisor' means set containment and 'greater' is also defined in terms of set containment.

3. Feb 22, 2014

HallsofIvy

Staff Emeritus
I thought "magma" had something to do with volcanos!

4. Feb 22, 2014

1MileCrash

The operation on the magma is union, so X is a divisor of A if there exists some set L in P(S) so that (X U L) = A.

I think this is equivalent to set containment (X is a subset of A).

I took "greater" to mean larger cardinality in this case. I reckon that the largest set that is a subset of A and B is (A intersect B), and that's why I think it is the GCD of A and B.

5. Feb 22, 2014

Dick

Ah ok, I see. So yes, L 'divides' A is equivalent to L is a subset of A. Defining "greater" in terms of cardinality can get you into some trouble with infinite sets. If the intersection of A and B is infinite, there may be many subsets with the same cardinality contained in it.

6. Feb 22, 2014

Dick

I had to look it up too. A "magma" is a set with a binary operation and that's it. No other properties necessary. Bourbaki invented the terminology.

7. Feb 22, 2014

1MileCrash

I didn't think of that.. perhaps instead of cardinality, I could say that the greatest common divisor is a superset of all other common divisors?

EDIT: Sorry about the terminology confusion, it is what my professors calls them, yes, a magma (M,*) is a set M with a binary operation *: M -> M.

When we talk about divisibility in a magma (M,*), * becomes analogous to multiplication and M becomes analagous to Z, regardless of what they actually are.

Last edited: Feb 22, 2014
8. Feb 22, 2014

Dick

Or just define A is 'greater' than B to mean B is a subset of A. Which amounts to the same thing.

Last edited: Feb 22, 2014
9. Feb 23, 2014

1MileCrash

Thank you for the assistance.