Difference between revisions of "Free monoid generated by"
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** {{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
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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
Contents
[hide]Definition
Given a set, X, there is a free monoid, (F,∗)[1].
- The elements of F are all the finite tuples, (x1,…,xn) (where xi∈X)
- The monoid operation (∗:F×F→F) is concatenation:
- ∗:((x1,…,xn),(y1,…,yn))↦(x1,…,xn,y1,…,yn)
- 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
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)
