Free monoid generated by

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

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].

• 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

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

Grade: D

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:

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

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

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