Difference between revisions of "Free monoid generated by"
From Maths
m (Added warning note) |
m |
||
| Line 10: | Line 10: | ||
** {{M|1=e=()}} - the "empty" tuple. | ** {{M|1=e=()}} - the "empty" tuple. | ||
The proof that this is indeed a monoid is below | The proof that this is indeed a monoid is below | ||
| + | ==Terminology== | ||
| + | The finite [[tuple|tuples]] of {{M|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}} | ||
| + | {{Requires references|grade=D|msg=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== | ==Examples== | ||
* This page can be considered an element of the monoid generated by the alphabet ([[union]] all the symbols too) | * This page can be considered an element of the monoid generated by the alphabet ([[union]] all the symbols too) | ||
Revision as of 14:00, 20 July 2016
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][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
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:
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
- Associativity is trivial
- Identity element being an identity element is trivial
(These might be good "low hanging fruit" for any newcomers)
