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Why in the world do they use the word "free" in "free monoid"? It driving me crazy to see where the "freedom" comes from.

Here is the Awodey's explanation of it, in terms of "baby lagebra" (sic.) but it is even more confusing:

A monoid M is freely generated by a subset A of M, if the following conditions hold

  1. Every element $m\in M$ can be written as a product of elements in A:
    $m = a_1 \cdot_{M} ... \cdot_{M} a_n, a_i\in A$
  2. No "nontrivial" relations hold in $M$, that is, if $a_1...a_j = a\prime_1 ... a\prime_k$, then this is required by the axioms for monoids.

to me this doesn't explain the word "free"...

Math level: novice

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You can think of "free" as meaning free of any equalities other than those implied by the monoid laws. So, e.g. in the free monoid generated by t the equality $\rm\:t\cdot 1\cdot t = t\:$ is true but $\rm\:t\cdot t = t\:$ is false, since the monoid laws imply the first but not the second equality. Monoids are not the best algebraic structure to begin learning about freeness because it is so trivial in this case. Instead, consider the polynomial ring R[x] as a free R-algebra. – Bill Dubuque Nov 13 '12 at 5:07
"polynomial ring R[x] as a free R-algebra" sorry that part went over my head. But I think I get what you are saying. It's weird, wouldn't free monoid then be just called "monoid" and anything else would be called "monoid with the following constraints"? – drozzy Nov 13 '12 at 5:11
@drozzy, It sounds like you're trying to learn category theory, but I would encourage you to first learn abstract algebra. Most of the developments in category theory come as natural generalizations of concepts one first learns in abstract algebra. – Christopher A. Wong Nov 13 '12 at 10:30
up vote 6 down vote accepted

You are an element of a monoid, say $x$. You want to strike out on your own, you want to act on another element $y$ and be a unique individual, not conforming to the laws of monoid society. But alas, the law says that the relation $xy = e$ holds; so when you act on $y$, you can't express yourself are only the identity element $e$ =(. You yearn for freedom,but you have been chained down by the tyranny of the monoid relation $xy = e$.

The above is an example of a monoid that is not free. Intuitively, a monoid is called free if, as you mentioned in your definition, there are no relations, i.e. equations, that relate the elements together, other than the conditions (axioms) that all monoids must obey. When there are relations, that means that the elements of the monoid must also obey additional constraints, and you can interpret this as being like a loss of freedom.

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Sorry for silly questions: 1. but by $x$ you mean monoid element and not element of the category right? 2. Also, when you say $xy$ it means $x\cdot y$? 3. If so, how can one element of monoid "act" on another? They can only be joined with a binary operations... or so I thought. – drozzy Nov 13 '12 at 5:06
Yes, I mean $x$ is an element of the monoid, and $xy = x \cdot y$, if $\cdot$ is the binary operation on the monoid. So by "act on", I just mean "apply binary operation with". – Christopher A. Wong Nov 13 '12 at 5:11
Oh ok cool! So the law that says $x\cdot y=e$ is some kind of "contrived" law made up by someone in addition to monoid laws? – drozzy Nov 13 '12 at 5:12
Well, it's not necessarily contrived, but yes, it's an additional constraint on top of the axioms that are absolutely required; presumably someone may have wanted these relations for some mathematical purpose. For example, consider the cyclic group $Z_p$, which is generated by some element $g$; then the relation that defines $Z_p$ is that $g^p = e$, the identity. This makes the group $Z_p$ not free, but I wouldn't exactly call the relation "contrived", but rather mathematically interesting! – Christopher A. Wong Nov 13 '12 at 5:34
Sorry, I am not familiar with cyclic groups or what generation of them means :-( But I am glad I am beginning to understand this stuff. What say you to my comment above to @Bill that it is rather weird to call these monoids "free" - since they seem like regular ol' monoids. Why not call the monoids with additional constraints "constrained monoids" (for example) and leave the regular monoids be? – drozzy Nov 13 '12 at 5:38

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