Talk:Free algebra

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 $$\rho$$, 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 $$\rho$$) on the set $$S$$ of free generators if, for every algebra $$\mathbf{B}$$ of type $$\rho$$ and function $$\tau : S \longrightarrow \mathbf{B}$$, there exists a unique homomorphism $$\psi :\mathbf{S} \longrightarrow \mathbf{B}$$ such that $$\psi \sigma = \tau$$.

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 $$\rho$$", 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)

Monoid ring
There is a mention in the text, but a closer integration with Monoid ring seems desirable. Deltahedron (talk) 21:18, 21 October 2012 (UTC)