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, $X$, there is a free monoid, $(F,*)$.

• The elements of $F$ are all the finite tuples, $(x_1,\ldots,x_n)$ (where $x_i\in X$)
• The monoid operation ($*:F\times F\rightarrow F$) is concatenation:
  • $*:((x_1,\ldots,x_n),(y_1,\ldots,y_n))\mapsto(x_1,\ldots,x_n,y_1,\ldots,y_n)$
• The identity element of the monoid is:
  • $e=()$ - the "empty" tuple.

The proof that this is indeed a monoid is below

Terminology

The finite tuples of $F$ 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

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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

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

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