Difference between revisions of "Free monoid generated by"
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| + | : 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== | ==Definition== | ||
Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}. | Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}. | ||
Revision as of 13:58, 20 July 2016
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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
Definition
Given a set, [ilmath]X[/ilmath], there is a free monoid, [ilmath](F,*)[/ilmath][1].
- The elements of [ilmath]F[/ilmath] are all the finite tuples, [ilmath](x_1,\ldots,x_n)[/ilmath] (where [ilmath]x_i\in X[/ilmath])
- The monoid operation ([ilmath]*:F\times F\rightarrow F[/ilmath]) is concatenation:
- [ilmath]*:((x_1,\ldots,x_n),(y_1,\ldots,y_n))\mapsto(x_1,\ldots,x_n,y_1,\ldots,y_n)[/ilmath]
- The identity element of the monoid is:
- [ilmath]e=()[/ilmath] - the "empty" tuple.
The proof that this is indeed a monoid is below
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)
