What is the Role of Formal Constructions in Algebraic Equivalence Classes?

[dmuthuk]

Hi, we often come across certain constructions in algebra that make use of some "formal" sum or "formal" linear combination or "formal" string of elements. Because this term is never defined, I have always been a little uncomfortable when it comes up. For a specific example, consider the construction of the free group on a set X. We begin by defining a "word" in X to be a formal string of elements in X. How do we make this a little more precise? Can we think of a word as an equivalence class of functions into X and concatenation as gluing these functions together? If so, how does that work?

 

[n_bourbaki]

A word is just a word. It is called formal purely because a priori X has no structure that allows us to construct words from the letters. And that is all that is going on. We just think of these things as if they made sense, when there is no innate structure, and then show that it makes sense.

 

[HallsofIvy]

If your elements are, say, a, b, c, then a "formal" word is just any combination of those letters. A "formal" sum of "abaca" and "bbac" might be "abacabbac" where the two words are combined in an obvious way.

 

[n_bourbaki]

I would have said the formal sum would be abaca+bbac, and an element in the group algebra. Again, we use the word formal simply because addition is not a naturally defined operation on strings.

 

[HallsofIvy]

I would have said the formal sum would be abaca+bbac, and an element in the group algebra. Again, we use the word formal simply because addition is not a naturally defined operation on strings.

That's probably better. I was assuming a specific operation without realizing it.

 

[dmuthuk]

Well, the reason I ask this question is because we sometimes treat sequences as functions from N into a set when it is intuitively obvious what we are talking about. Is this unecessary precision or are there situations where intuition can be misleading? For example, in Dummit and Foote, the free R-module F(A) on a set A is constructed by specifiying the elements of F(A) to be the collection of all set functions f : A --> R with finite support. I guess this makes precise the notion of a "formal" R-linear combination.