Non Integer Exponents for Cartesian Products

In summary: I'm not sure if that would be meaningful or useful.Anyway, I would be grateful if anyone could tell me if any of this exists already, is complete nonsense, or which area of mathematics it would belong to.
  • #1
Daron
15
0
I know the Cartesian product for an algebraic structure: A x B = {(a,b): a ∈ A, b ∈ B}

Which naturally gives An = {(a1, a2, ... , an): ai ∈ A ∀ i}

Some of the time, at least we can also have a non integer n.
For example [A x A x A]2/3 = A x A.

Is there any way of continuing the exponent n into non integer numbers for any exponent? It might involve creating an algebraic structure very unlike the original A, but which is still isomorphic when raised to an appropriate power, with respect to the Cartesian product.

I have another question as well:

Is there a way to define a group, which has as elements all groups, or all algebraic structures of a certain kind? Could this be made into a ring? If it could, perhaps we could create a structure exponential using the two operations. A metric might also be required on the set of all structures to make sure the series converges.

Anyway, I would be grateful if anyone could tell me if any of this exists already, is complete nonsense, or which area of mathematics it would belong to.
 
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  • #2
Daron said:
I know the Cartesian product for an algebraic structure: A x B = {(a,b): a ∈ A, b ∈ B}

Which naturally gives An = {(a1, a2, ... , an): ai ∈ A ∀ i}

Some of the time, at least we can also have a non integer n.
For example [A x A x A]2/3 = A x A.

Is there any way of continuing the exponent n into non integer numbers for any exponent? It might involve creating an algebraic structure very unlike the original A, but which is still isomorphic when raised to an appropriate power, with respect to the Cartesian product.
I don't think there's any way to do this without creating some sort of mess. You will already run into a problem if you try to define the square root of a group to be a group. Indeed, there are examples of nonisomorphic groups A and B such that [itex]A^2 \cong B^2[/itex]. This means that "square roots" of isomorphic things won't have to be isomorphic!

I don't know what kind of structure you would ask the "square root" of a specific algebraic object to be...

I have another question as well:

Is there a way to define a group, which has as elements all groups, or all algebraic structures of a certain kind? Could this be made into a ring? If it could, perhaps we could create a structure exponential using the two operations. A metric might also be required on the set of all structures to make sure the series converges.
Due to set-theoretic difficulties, the answer to your first question is no for groups (and many other types of algebraic structures). This is because any nonempty set can be given a binary operation that turns it into a group. (This assertion is equivalent to the axiom of choice.) Thus your group of all groups will in particular be a set of all (nonempty) sets, which is problematic.
 
  • #3
Hey Daron and welcome to the forums.

This is a very interesting question.

It seems that you could attack this from two ways.

The first one is cardinality. Let's say we have a set A that has 10 elements. We know that A^2 has 100 element pairs. Now somewhere in-between A and A^2 you would have somewhere between 10 and 100 elements.

The elements would have to obey a specific permutation law that goes hand in hand with the ordered definition of elements in the set. So for example if my set was {0,1,2,3,4,5,6,7,8,9} then we would expect permutations involving the lower numbers would appear before those involved with higher numbers in terms of the first element of the pair (i.e. pairs (1,x) will come before (9,x)).

The next thing to address is how one would handle countable and uncountable sets.

If you had a countable set like my example above, you would need to use a function like say a floor function to get the right number of elements to permute and then use the standard permutation function to get your output set.

In order to find the number of elements it would have to be a derivative of the log function in a given base since we are dealing with systems involving exponents.

Basically every order of magnitude will add an extra dimension to the object of the final set element. The easiest way to picture this is to think a normal number in a given base: we can represent 10 and 98 with two digits with 101 needs three digits and each digit corresponds to representing a sub-element as part of a final element in your final set.

I think I could derive a formula to do this for countable sets in terms of logs, floors and/or ceilings and a standard permutation function to generate the right set that gives the right answers for integer exponents and the "interpolated" versions thereof if you are interested.
 

Related to Non Integer Exponents for Cartesian Products

What are non-integer exponents for Cartesian products?

Non-integer exponents for Cartesian products refer to the use of fractional or decimal numbers as the power or exponent in a Cartesian product. This allows for more precise and accurate calculations when dealing with the dimensions of a Cartesian product.

How are non-integer exponents used in Cartesian products?

Non-integer exponents are used in Cartesian products to represent the number of times a set is multiplied by itself. For example, a Cartesian product with an exponent of 1.5 would mean that the set is multiplied by itself 1.5 times, resulting in a product with a larger number of elements.

What are the benefits of using non-integer exponents in Cartesian products?

Using non-integer exponents in Cartesian products allows for more precise and accurate calculations, as well as the ability to represent and manipulate fractional dimensions. This can be particularly useful in fields such as engineering and physics where precise measurements are crucial.

What are some common applications of non-integer exponents in Cartesian products?

Non-integer exponents in Cartesian products are commonly used in fields such as geometry, calculus, and statistics. They can also be applied in computer science for data analysis and algorithm design.

Are there any limitations or restrictions when using non-integer exponents in Cartesian products?

While non-integer exponents can provide more precise calculations, there may be limitations or restrictions when working with very large or very small fractional exponents. Additionally, some mathematical operations may not be defined for certain non-integer exponents, so it is important to carefully consider the context and purpose of using these exponents in Cartesian products.

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