What criteria identifies a math operation as a "product"?

[ibkev]

Today I was reading about geometric algebra and a kind of vector product that combines the dot and cross/wedge products together and it got me thinking about the meaning of "product". My math background is from an engineering perspective and I've always just accepted the dot and cross products as useful functions but it occurred to me that if someone had not told me that the dot and cross operators were "products" I wouldn't know any criteria that would allow me to justify calling them as such.

I don't necessarily mean vectors specifically either. Imagine you were defining some new abstract mathematical object, say a foobar, how would you be able to tell which of the functions you came up with for working with them is a "product" as opposed to some other operation?

Also, what math discipline would teach me this? Abstract algebra maybe?

 

[mfb]

I don't think there is a formal definition, people go by what is useful.
At the very least it should be an operation that takes two elements of a set. Often it produces another element of the same set (product of real numbers, product of matrices, vector product, ...) but it doesn't have to: The scalar product takes vectors and returns a real or complex number. Usually it is linear, I don't find a counterexample to that.

 

[PeroK]

If you only have one binary operation, then generally there is no clear distiction between "addition" and "multiplication". That said, addition is generally commutative, so if the operation is non-commutative, then it's a "product".

When you have two binary operations, multiplication is distributive over addition:
$$a \cdot (b + c) = (a \cdot b) + (a \cdot c)$$
For these reasons, the vector dot and cross products are "product" operations. The cross product is non-commutative and both distribute over vector addition.

 

[ibkev]

Ah - the definition is more about the way it can be used than what it actually is - thanks!

 

[ibkev]

BTW, if I had taken a math degree, is it abstract algebra that would have taught me that?

 

[PeroK]

BTW, if I had taken a math degree, is it abstract algebra that would have taught me that?

I'd never thought of that question before. I did a maths degree because I can work stuff like that out for myself! The same way an engineer can see how something works without being taught; or a music student understands how a tune works.

 

[robphy]

Possibly useful (but I am not qualified to explain the distinction... maybe someone more qualified can do so):

https://bartoszmilewski.com/2015/01/07/products-and-coproducts/
bolding mine...

Asymmetry

We’ve seen two set of dual definitions: The definition of a terminal object can be obtained from the definition of the initial object by reversing the direction of arrows; in a similar way, the definition of the coproduct can be obtained from that of the product. Yet in the category of sets the initial object is very different from the final object, and coproduct is very different from product. We’ll see later that product behaves like multiplication, with the terminal object playing the role of one; whereas coproduct behaves more like the sum, with the initial object playing the role of zero. In particular, for finite sets, the size of the product is the product of the sizes of individual sets, and the size of the coproduct is the sum of the sizes.

https://unapologetic.wordpress.com/2007/06/11/products-and-coproducts/
http://blog.higher-order.com/blog/2014/03/19/monoid-morphisms-products-coproducts/

https://en.wikipedia.org/wiki/Product_(category_theory)
https://en.wikipedia.org/wiki/Coproduct

https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Products_and_Coproducts

https://mathworld.wolfram.com/CategoryProduct.html
https://mathworld.wolfram.com/Coproduct.html

 

[ibkev]

Oh interesting! Category Theory is something that I’ve never looked into before. Maybe that would make for a fun project. :) Thanks!

 

[mathwonk]

My first acquaintance with a general "product", as an operation, not as a construction from component objects as in category theory, was in Dieudonne' 's Foundations of modern analysis, where he proves the "product rule" for differentiation of various products of functions. He uses only the property that the "product" is distributive over addition, and deduces the usual product rule for the derivative, which then applies to dot products as well as cross products, and other products, of functions. In category theory, a product is a construction for forming new objects from a family of old ones, not at all the same idea I think, as a product operation on a given object, but I could be wrong (as so often before).

 

[Mark44]

it got me thinking about the meaning of "product".

A product is usually associated with some kind of multiplication, something that @robphy highlighted in the quote in his post.

 

[ibkev]

A product is usually associated with some kind of multiplication, something that @robphy highlighted in the quote in his post.

I guess that’s the reason for my question. Exploring if there is a difference between multiplying two objects together and taking their product. Looking at the formula for a cross product, to me it’s not obviously a means of multiplying two vectors because there’s no associated divide and it’s anticommutative. It’s certainly useful but not conformant to intuition based on multiplying scalars and complex numbers.

What I took from PeroK’s post above is that I’d come to associate with products some specific characteristics of multiplying scalars/complex numbers that don’t hold in general.

Does that sound like a reasonable conclusion?

 

[FactChecker]

I think that you are right in referencing abstract algebra as a good source for deciding if an operation should be called "multiplication". You already have the operation of addition and you would want your second operation to have the properties required for a ring or pseudo-ring (see this). You would want the operation to be binary, closed, associative, have the left and right distributive property over the addition operation, and (usually) have a multiplicative identity element, I.

 

[jbriggs444]

Looking at the formula for a cross product, to me it’s not obviously a means of multiplying two vectors because there’s no associated divide and it’s anticommutative.

If we represent scalar multiplication using juxtaposition and the cross product with $\times$ then the cross product upholds:
$$2\vec A \times 2 \vec B = 4(\vec A \times \vec B)$$That is something that multiplication does but which addition does not$$2\vec A + 2 \vec B \ne 4(\vec A+\vec B)$$ So we think of cross product as a form of multiplication.

 

[zinq]

The statement "A product is associated with some kind of multiplication" is true, but that doesn't really explain anything.

As others have said, when there's only one binary operation — notably when you have a group — it's often termed "addition" if that operation is commutative, and termed "multiplication" if that operation is non-commutative.

When there are two binary operations and one of them is commutative, that one is usually termed "addition" and — if the other one distributes over the first one — that one is called multiplication. (Even when the second one happens to also be commutative.)

Common examples include vector spaces (where the "multiplication" is just scalar multiplication); fields — such as the rational numbers, the reals R, or the complexes C; rings — such as the integers, or the set of all n×n matrices M_n(K) with entries in a field K; and hybrid creatures like the quaternions H (sometimes called a skew-field) or the octonions O (which are non-associative). Yet another example is tensor "product" of vector spaces, which distributes over direct sum of vector spaces (up to isomorphism of vector spaces).

 

[ibkev]

Thanks for all the replies folks. I dug up a copy of “A Book of Abstract Algebra” by Pinter that seems unusually readable for a math text, so I think I have my next beach read! :)