Talk:Free algebra

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Mathematics rating: Stub Class Mid Priority  Field: Algebra

[edit] Universal algebra

In articles like this, I understand why the category theory definition is nice, as it is so general, but I don't (personally) find it very useful. A definition in a universal algrebra book or paper would look more like this:

Let S be any set, let \mathbf{A} be a algebra of type ρ, and let \psi :S \longrightarrow \mathbf{A} be a function. we say that (\mathbf{A}, \psi) (or informally just \mathbf{A}) is a free algebra (of type ρ) on the set S of free generators if, for every algebra \mathbf{B} of type ρ and function \tau : S \longrightarrow \mathbf{B}, there exists a unique homomorphism \psi :\mathbf{S} \longrightarrow \mathbf{B} such that ψσ = τ.

So I have a couple questions:

1) Is there a central WP place where the benifits of catagory theory type definitions of concepts are weighed, and from which I could judge when other perspectives are appropriate?

2) Assuming this definition is not horribly mangled, would it be appropriate to add a universal algebra type definition of the a free algebra such as this one to this article?

I am assuming this discussion already exists somewhere on some article, and I don't want to have it all over again. Thanks. Smmurphy(Talk) 23:20, 21 February 2006 (UTC)

1) Probably best to as on WP:WPM.
2) If you can incorporate it clearly into what's already there, then, yes. However, I have no idea of what you mean by "of type ρ", and thus this would need to be expanded upon first. linas 17:19, 11 February 2007 (UTC)
By type of an algebra I meant the arity of the operations of the algebra, which is an important part of how the algebra is defined in a sort of Universal Algebra sense. I had hoped that after adding this question, someone would look at the definition I gave and give it some criticism (thereby helping me understand something I needed to know at the time - ; ) too late now) as well as clear up the question. I'll mention it over at the project page now. Thanks. Smmurphy(Talk) 02:07, 13 February 2007 (UTC)
The general UA definition is covered in some detail at free object, and this specific definition has been added to a section there by Hans Adler. JackSchmidt (talk) 13:48, 23 April 2008 (UTC)