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)