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

Definition

Given a set, [ilmath]X[/ilmath], there is a free monoid, [ilmath](F,*)[/ilmath].

• 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

1. Associativity is trivial
2. Identity element being an identity element is trivial

(These might be good "low hanging fruit" for any newcomers)