# Talk:Free algebra

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

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 ${\displaystyle \rho }$", 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)