What is the GCD of A and B on a Union Magma?

[1MileCrash]

Homework Statement

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?

Homework Equations

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?

 

[Dick]

Homework Statement

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?

Homework Equations

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?

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.

 

[HallsofIvy]

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

 

[1MileCrash]

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.

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.

 

[Dick]

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.

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.

 

[Dick]

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

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.

 

[1MileCrash]

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.

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.

 

[Dick]

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?

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

 

[1MileCrash]

Thank you for the assistance.