How Can V Be Transformed into Different Algebraic Structures?

[Dragonfall]

What sort of operations can you pair up with V (class of all sets in ZFC) that turns it into various algebraic structures? For example, symmetric difference turns it into a group.

Now you must wave your hands a little since the algebraic structures are technically sets, and so are the functions, etc.

 

[enigmahunter]

For instance, a forgetful functor $$U:R Mod \rightarrow Set$$ has a left adjoint F such that $$X \mapsto FX$$, which generates a free R-module using a basis set. So a free construction generates an algebraic structure using a set.

http://en.wikipedia.org/wiki/Forgetful_functor" is the description from wiki,

"As this a fundamental example of adjoints, we spell it out: adjointness means that given a set X and an object (say, an R-module) M, maps of sets correspond to maps of modules : every map of sets yields a map of modules, and every map of modules comes from a map of sets."

$$Hom_{Mod R}(Free_{R}(X), M) = Hom_{Set}(X, Forget(M))$$

 

[Dragonfall]

How would you turn V into a field?

 

[enigmahunter]

How would you turn V into a field?

I think a http://en.wikipedia.org/wiki/Construction_of_the_real_numbers" would do.

"A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers."

As you know, rational numbers can be constructed using integers and integers can be constructed using natural numbers that can be constructed using sets from constructivists' view. For above situations, the operators for algebraic structures can be converted into set operators.