Formal Construction in Algebra: Exploring Words & Equivalence Classes

In summary: But is this really necessary? It seems like the collection of all set functions could also be thought of as a "formal" R-linear combination. In summary, the formal linear combination is a way to make a collection of elements more precise.
  • #1
dmuthuk
41
1
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?
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
n_bourbaki said:
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.
 
  • #6
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.
 
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1. What is formal construction in algebra?

Formal construction in algebra is a method of representing mathematical concepts and operations using symbols, equations, and logical rules. It involves working with abstract symbols rather than specific numbers or quantities.

2. What is the purpose of exploring words in formal construction?

Exploring words in formal construction allows us to understand how mathematical concepts and operations can be represented using symbols. It helps us develop a deeper understanding of algebraic language and its application in problem-solving.

3. What are equivalence classes in formal construction?

Equivalence classes in formal construction refer to groups of symbols or expressions that have the same meaning or value. It is a way of categorizing and organizing different algebraic representations that are mathematically equivalent.

4. How do equivalence classes relate to algebraic operations?

Equivalence classes are important in algebraic operations because they allow us to see how different algebraic expressions can be used to represent the same mathematical concept. This helps us to manipulate and solve equations more efficiently.

5. How can formal construction in algebra be applied in real life?

Formal construction in algebra has many practical applications in fields such as engineering, computer science, and economics. It allows us to model real-world problems using equations and symbols, making complex calculations easier and more efficient.

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