Sets and functions that gain more structure with context

[ProfuselyQuarky]

So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it doesn't make any sense to think that ##f## is group homomorphism.

I started to briefly learn about rings and how every field is a ring, every ring is a group, which makes every field a group, too. So what if sets ##A## and ##B## have binary operations? In this case, they might be groups! If ##
A## and ##B## are, indeed, groups, then we can question whether ##f## is group homomorphism, too.

Whether sets and functions have more structure or not is completely dependent on the context which is being spoken of. To me, that just doesn't sound right. Shouldn't there be a precise way of determining whether a set or function has more structure or not? Is there?

 

[micromass]

So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it doesn't make any sense to think that ##f## is group homomorphism.

I started to briefly learn about rings and how every field is a ring, every ring is a group, which makes every field a group, too. So what if sets ##A## and ##B## have binary operations? In this case, they might be groups! If ##
A## and ##B## are, indeed, groups, then we can question whether ##f## is group homomorphism, too.

Whether sets and functions have more structure or not is completely dependent on the context which is being spoken of. To me, that just doesn't sound right. Shouldn't there be a precise way of determining whether a set or function has more structure or not? Is there?

Oh boy, I'm going to have to resist speaking of categories and forgetful functors, since that answers your question directly. But I'll take a different approach.

What you do is to see a set as distinct from a structure like a group. This is certainly the way to think about it, but I think you want something more formal.

So what is a group? Formally, a group is an ordered pair ##\mathcal{G} = (G,*)## where ##*:G\times G\rightarrow G## is a function. Note that ordered pairs are defined as ##(a,b) = \{\{a\},\{a,b\}\}##.
What I just did is make a distinction between the group ##\mathcal{G}## and the underlying set ##G##. Both are sets, but somewhat different sets.

Let's take a look at functions. What is a function pure formally? A function ##G\rightarrow H## is defined as an ordered tuple ##f=(G,H,R)## where ##R\subseteq G\times H## with the property that for all ##g\in G##, there is a unique ##h\in H## such that ##(g,h)\in R##. We then use the notation ##f(g)=h##. So a function is a set too.

Now for group homomorphisms. A group homomorphism ##F:\mathcal{G}\rightarrow \mathcal{H}## can be defined as ##F = (\mathcal{G},\mathcal{H},f)## with ##f:G\rightarrow H## the underlying set function. So now there is a very clear difference between a group homomorphism and a usual function. This distincition is never made (for good reasons, it's really tedious) in texts.

 

[fresh_42]

Oh boy, I'm going to have to resist speaking of categories and forgetful functors ...

Lol, that was a good one. I first thought you would say "... resist on giving a lecture ..." but this was even better!

 

[PeroK]

Whether sets and functions have more structure or not is completely dependent on the context which is being spoken of. To me, that just doesn't sound right. Shouldn't there be a precise way of determining whether a set or function has more structure or not? Is there?

Where would you stop? Suppose ##A## and ##B## are both ##\mathbb{R}##. They are both groups, rings, fields, metric spaces, topological spaces, vector spaces ... Any number of binary operations or metrics or topologies can be defined on them. You are free to invent your own. ##f## could be a continuous or not depending on the metric or topology chosen; and, integrable and/or differentiable

In a nutshell, context is everything. There isn't something uniquely and exclusively well-defined called ##\mathbb{R}##. It has as much or as little structure as the context requires. We all talk loosely about "the real numbers" or "the sine function", but without a context they aren't actually well-defined.

 

[fresh_42]

One can regard almost everything as a set. Probably everything if one doesn't care whether it makes sense somehow. It is plainly the basic concept that allows to collect things and distinguish them from other things that do not belong to the set.
A function is already a constraint, since it is a special relation: you must not have two images of the same element.

Therefore the variety of sets is vast and often we are only interested in sets, that satisfy some properties. You mentioned groups, rings and fields. All of them help us to establish frameworks to solve equations, calculate orbits and so on. If you want to determine the solution of ##x^2 - 4 = 0## it doesn't help to allow a set of stones in the consideration. I wanted to give a mathematical example of a set that isn't of interest rather than stones but this turned out to be difficult since, e.g. a set of open sets may also contribute to the problem, simply from a different angle.
So the additional structures on sets simply bring order to the chaos.
The same with functions. In connection to given structures one usually requires them to preserve those structures. To allow any functions will simply be too many and of little help.

 

[ProfuselyQuarky]

Oh boy, I'm going to have to resist speaking of categories and forgetful functors

Good idea (but if you care to elaborate for others, go for it).

What you do is to see a set as distinct from a structure like a group. This is certainly the way to think about it, but I think you want something more formal.

Never thought about it that way before. Since a structure is said to be an "arbitrary set", I've thought of it as almost a set underlying the "main" set--and not two distinct things.

So what is a group? Formally, a group is an ordered pair ##\mathcal{G} = (G,*)## where ##*:G\times G\rightarrow G## is a function. Note that ordered pairs are defined as ##(a,b) = \{\{a\},\{a,b\}\}##.
What I just did is make a distinction between the group ##\mathcal{G}## and the underlying set ##G##. Both are sets, but somewhat different sets.

Let's take a look at functions. What is a function pure formally? A function ##G\rightarrow H## is defined as an ordered tuple ##f=(G,H,R)## where ##R\subseteq G\times H## with the property that for all ##g\in G##, there is a unique ##h\in H## such that ##(g,h)\in R##. We then use the notation ##f(g)=h##. So a function is a set too.

Now for group homomorphisms. A group homomorphism ##F:\mathcal{G}\rightarrow \mathcal{H}## can be defined as ##F = (\mathcal{G},\mathcal{H},f)## with ##f:G\rightarrow H## the underlying set function. So now there is a very clear difference between a group homomorphism and a usual function. This distincition is never made (for good reasons, it's really tedious) in texts.

Thanks, I see what you're getting at . I wish I could connect these definitions easier than I am. If the difference between a group homomorphism and a usual function is so apparent, what is so tedious about the distinction?

Where would you stop?

Good question. I'm not sure. But I think real numbers do have a set definition, even without context. If I just randomly chose any number, I'd know whether it's real or not without the help of any other information.

I wanted to give a mathematical example of a set that isn't of interest rather than stones but this turned out to be difficult since, e.g. a set of open sets may also contribute to the problem, simply from a different angle.

If nearly everything can be regarded as a set, how is it so hard to find a mathematical example? Surely there must be a simple one amidst them all?

So the additional structures on sets simply brings order to the chaos.
The same with functions. In connection to given structures one usually requires them to preserve those structures. To allow any functions will simply be too many and of little help.

Lovely quote!

 

[fresh_42]

If nearly everything can be regarded as a set, how is it so hard to find a mathematical example? Surely there must be a simple one amidst them all?

Well, let me try. Set of probabilities should be pretty far away from ##x^2 - 4 = 0##. But then you could think of probabilistic algorithms to solve those equations. Topological sets? Nope: the equation already defines a closed set. Algebra can be ruled out as a whole since we would end up with Galois theory which involves groups and fields. The same for number theory. Combinatorics? Bad idea. Binomial coefficients are ##\begin{pmatrix}n \\ k\end{pmatrix}##. Set theory and logic? That contains all other mathematics automatically, so especially our equation.

Those have been my thoughts before I chose the stones instead. The risk someone finding a bridge was simply to high.

 

[ProfuselyQuarky]

Well, let me try. Set of probabilities should be pretty far away from ##x^2 - 4 = 0##. But then you could think of probabilistic algorithms to solve those equations. Topological sets? Nope: the equation already defines a closed set. Algebra can be ruled out as a whole since we would end up with Galois theory which involves groups and fields. The same for number theory. Combinatorics? Bad idea. Binomial coefficients are ##\begin{pmatrix}n \\ k\end{pmatrix}##. Set theory and logic? That contains all other mathematics automatically, so especially our equation.

Those have been my thoughts before I chose the stones instead. The risk someone finding a bridge was simply to high.

That makes sense. Thanks

 

[The Bill]

There are many types of structures that can be added to sets. However, given a specific structure such as continuous group structure, it is possible that not all sets can admit that structure in the way you'd like.

If you think of operating on the left(or right, but I'll stick with left for this discussion) by a particular element as a function from a group G into itself, not all continuous sets can have a group structure where those functions of operating on the left are all continuous.

For example, think of the points on a circle as a set. We can give the circle a group structure based on rotation. You can parametrize this as the unit circle with addition of angles as the group operation, while identifying angles which differ by a full rotation as the same point. Another way to think of it is as the unit circle centered on 0 in the complex number plane. The function of rotating all the points in a circle by an amount given by the complex angle of a single given point is continuous.

So, the circle can be given the structure of a continuous group. So can the real number line with addition as the group operation. Cartesian products of these can also be continuous groups, with the simplest operation just being the above operations operating separately on each part. So, the surface of a cylinder is the Cartesian product of the real line and a circle, and a simple group operation is to give each point an angle coordinate on the circle and a "height" coordinate on the line which operate separately by addition like in the circle and line. Similarly, the surface of a doughnut can be the Cartesian product of two circles.

However, the points on a sphere cannot be made into a continuous group. Try any way you like, there's no way to do it while including every point on the sphere as a group element.

So, it's not just the context. Sometimes, a particular set just can't admit a certain structure. A simpler example is that we have to remove 0 from the real number line to make a group over multiplication.

In case you're interested, Naive Lie Theory by John Stillwell is an excellent introduction to continuous groups, specifically.

 

[ProfuselyQuarky]

There are many types of structures that can be added to sets. However, given a specific structure such as continuous group structure, it is possible that not all sets can admit that structure in the way you'd like.

If you think of operating on the left(or right, but I'll stick with left for this discussion) by a particular element as a function from a group G into itself, not all continuous sets can have a group structure where those functions of operating on the left are all continuous.

For example, think of the points on a circle as a set. We can give the circle a group structure based on rotation. You can parametrize this as the unit circle with addition of angles as the group operation, while identifying angles which differ by a full rotation as the same point. Another way to think of it is as the unit circle centered on 0 in the complex number plane. The function of rotating all the points in a circle by an amount given by the complex angle of a single given point is continuous.

So, the circle can be given the structure of a continuous group. So can the real number line with addition as the group operation. Cartesian products of these can also be continuous groups, with the simplest operation just being the above operations operating separately on each part. So, the surface of a cylinder is the Cartesian product of the real line and a circle, and a simple group operation is to give each point an angle coordinate on the circle and a "height" coordinate on the line which operate separately by addition like in the circle and line. Similarly, the surface of a doughnut can be the Cartesian product of two circles.

However, the points on a sphere cannot be made into a continuous group. Try any way you like, there's no way to do it while including every point on the sphere as a group element.

So, it's not just the context. Sometimes, a particular set just can't admit a certain structure. A simpler example is that we have to remove 0 from the real number line to make a group over multiplication.

Wow, that's a lot of information. Thanks, I'm just trying to process your post to its entirety :)

In case you're interested, Naive Lie Theory by John Stillwell is an excellent introduction to continuous groups, specifically.

Thanks for the suggestion. I can only manage one book at a time, but will definitely look into it.