A story reminiscent of something

Question

An old Russian mathematician once told me this strange story...

“In a galaxy far away there is a civilized species.

The skin colors of everyone in the species is blue or green. Among the species, there are many different ethnicities of people who each live on many different planets. Due to each people groups’ ideology of racial supremacy, they each have a law that says that everyone must use a special device to more-purely reproduce and only with those on their own planet so that the child remains of the same ethnicity.

This reproduction device has two chambers and both must be filled to work. One chamber must only be occupied by those who are blue and the other chamber must only be occupied by those who are green. The device clones each person’s DNA and from it produces a third person of the same ethnicity.

A real-case scenario once followed:

A cloned with B producing C.
C cloned with D producing E.
Then B cloned with D producing F.
Then A cloned with F producing an exact copy of E.

One day, one of the Es discovered a portal removing him to a higher dimension and on his return he had become $\exists$.

E cloned with $\exists$ and produced a special person, 0, who contained no DNA.

Now anybody who cloned with 0 produced an exact replica of themselves.

Soon everybody performed this process of creating a double to enter the portal in order to reproduce more purely.

So to recap, there are a bunch of people groups of this species. And they each have a law to ensure purer DNA by using a special device for reproduction. And they have discovered some interesting properties that happen when special people enter this device.”

After finally finishing this bizarre story, the mathematician asked in his heavy Russian accent,

“Now suppose there are a billion groups of this species under this law. As racist as these groups are, do you think that the people in them fight because of encountering differing skin colors? Why or why not?”

I think I may have misinterpreted a word because of his accent. But it all seems so random anyway! How am I supposed to get a logical answer?

Answer 1

The story describes

a group in mathematics.

The skin colors of everyone in the species is blue or green.

Not sure about this one right now. This might refer to a special kind of group(oid?) that only accepts pairs where one element comes from subset B and the other element comes from the remainder G, but possibly this just suggests that order does actually matter for the machine; the group isn't necessarily Abelian.

Among the species, there are many different ethnicities of people who each live on many different planets.

There are many kinds of groups, with different behaviors when endowed with their operation.

Due to each people groups’ ideology of racial supremacy, they each have a law that says that everyone must use a special device to more-purely reproduce and only with those on their own planet so that the child remains of the same ethnicity.

This describes the closure condition of a group: the group operation is only defined by pairs of elements of a group, and results in another element that is the same group.

This reproduction device has two chambers and both must be filled to work. One chamber must only be occupied by those who are blue and the other chamber must only be occupied by those who are green. The device clones each person’s DNA and from it produces a third person of the same ethnicity.

This further emphasizes closure. It also suggests that skin color determines whether an element is a left or right element, so this might be a special kind of group.

A real-case scenario once followed:

A cloned with B producing C.

C cloned with D producing E.

Then B cloned with D producing F.

Then A cloned with F producing an exact copy of E.

This describes the associativity condition: A@B = C, C@D = E, so substituting we have (A@B)@D = E. Likewise B@D = F, A@F = E, so we have A@(B@D) = E. So A@(B@D) = (A@B)@D.

One day, one of the Es discovered a portal removing him to a higher dimension and on his return he had become ∃.

E cloned with ∃ and produced a special person, 0, who contained no DNA.

Now anybody who cloned with 0 produced an exact replica of themselves.

The first sentence strongly implies that E and ∃ are inverses, to be confirmed in the next two lines. The second sentence states that E@∃ = 0, which initially doesn't mean much, but the third sentence is a strong condition that states that 0@x = x for any x. Therefore 0 is our identity element (which is the last of the four group conditions), and confirms that the special portal creates an inverse element.

Soon many other planets performed this same process to create a 0 in order to reproduce more purely.

This is the actual invertability condition - every element has an inverse.

So to recap, there are a bunch of people groups of this species. And they each have a law to ensure purer DNA by using a special device for reproduction. And they have discovered some interesting properties that happen when special people enter this device.

So to recap, there are many groups in mathematics. Each group has a group operation taking pairs of elements to elements. And groups and their operation satisfy some interesting properties.

“Now suppose there are a billion groups of this species under this law. As racist as this species is, do you think that they fight because of encountering differing skin colors? Even among their own communities? Why or why not?”

To the best of my knowledge, skin colors don't matter at all, but are just a way to show that the function doesn't commute. I'm not sure what the word "fight" really refers to though.

Answer 2

This is as far as I understood:

reproduce and only with those on their own planet so that the child remains of the same ethnicity

I think this means Closure property

A cloned with B producing C. C cloned with D producing E. Then B cloned with D producing F. Then A cloned with F producing an exact copy of E

This seems to be associative property

E cloned with ∃ and produced a special person, 0, who contained no DNA. Now anybody who cloned with 0 produced an exact replica of themselves

This is an inverse with 0 as identity, like an additive inverse with 0 as identity

I think

Each group of species is a universal set and each member is a subset of the universal set. (The inverse is Complementary set). And the fight refers to intersection of the sets. So fights/intersection of subsets of same universal set result in another subset of same universal set. But fights/intersections of subsets of different universal sets result in Null set. So it can be argued that Null set means no fight.