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
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
Notation
- We often identify [ilmath]x\in X[/ilmath] with [ilmath](x)\in F[/ilmath], and singletons of [ilmath]F[/ilmath] (ie: [ilmath](y)\in F[/ilmath] with [ilmath]y\in X[/ilmath].
- This notation extends further, and (especially in the case of the free semigroup generated by [ilmath]X[/ilmath]^{[Note 1]}) we write [ilmath](x_1,x_2,\ldots,x_{n-1},x_n)[/ilmath] as a product or word, [ilmath]x_1x_2\ldots x_{n-1}x_n[/ilmath]
Terminology
- The finite tuples of [ilmath]F[/ilmath] 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 [ilmath]F[/ilmath] written in the "product" notation, [ilmath]x_1\ldots x_n[/ilmath]. The reference^{[1]} implies this.
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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
- ↑ 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 [ilmath]X[/ilmath]), 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.
References
- ↑ ^{1.0} ^{1.1} Abstract Algebra - Pierre Antoine Grillet