View source for Free monoid generated by

{{Stub page|grade=A|msg=Demote once fleshed out and minimally complete}}
: Be sure to check [[Discussion of the free monoid and free semigroup generated by a set]], as there are some things to note
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==Definition==
Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}. 
* The elements of {{M|F}} are all the finite [[tuple|tuples]], {{M|(x_1,\ldots,x_n)}} (where {{M|x_i\in X}})
* The monoid operation ({{M|*:F\times F\rightarrow F}}) is concatenation:
** {{M|*:((x_1,\ldots,x_n),(y_1,\ldots,y_n))\mapsto(x_1,\ldots,x_n,y_1,\ldots,y_n)}}
* The identity element of the monoid is:
** {{M|1=e=()}} - the "empty" tuple.
The proof that this is indeed a monoid is below
===Notation===
* We often identify {{M|x\in X}} with {{M|(x)\in F}}, and singletons of {{M|F}} (ie: {{M|(y)\in F}} with {{M|y\in X}}.
* This notation extends further, and (especially in the case of the [[free semigroup generated by]] {{M|X}}<ref group="Note">We do this because the semigroup has no identity (in fact, is considered as the set of all tuples of length greater than or equal to one of elements of {{M|X}}), it has no "empty tuple", writing an empty tuple as a "word" would be an empty word! You couldn't even tell it was there.</ref>) we write {{M|(x_1,x_2,\ldots,x_{n-1},x_n)}} as a product or ''word'', {{M|x_1x_2\ldots x_{n-1}x_n}}

==Terminology==
* The finite [[tuple|tuples]] of {{M|F}} are sometimes called "words".
** {{Warning|The "word" terminology may be specific to the [[free group]], however I wouldn't be surprised if word is used in this context too, so I deem it still worth mentioning}}
** {{Caution|Word may only be used for elements of {{M|F}} written in the "product" notation, {{M|x_1\ldots x_n}}. The reference<ref name="AAPAG"/> implies this.}}
{{Requires references|grade=D|msg=While not explicitly said, the main reference doesn't deal with these objects in great detail, however usually such tuples are called words, at least with free groups (see warning)}}
==Examples==
* This page can be considered an element of the monoid generated by the alphabet ([[union]] all the symbols too)
==Proof that this is indeed a monoid==
# Associativity is trivial
# Identity element being an identity element is trivial
(These might be good "low hanging fruit" for any newcomers)
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Definition|Monoid Theory|Abstract Algebra}}