then we may improperly say that is a word on , considering simply as the free monoid on .
The word is called reduced when for each .
Now, starting from a word , we can iteratively erase factors from whenever , and this iterative process, that we call reduction of , produce a reduced word . At each step of the process there may be more than one couple of adiacent letters candidate to be erased, so we may ask if different sequences of erasing may produce different reduced words. The following theorem answers the question.
Each couple of reduction of a same word produce the same reduced word .
The unique reduced word is called the reduced form of . Thus there exists a well define map that send a word to his reduced form .
We can use the map to build the free group on in the following way. Let be the set of reduced words on , i.e.
i.e. is the reduced form of the juxtaposition of the words and . The we have the following result.
It is well known from universal algebra that is unique up to isomorphisms. With this construction, the map [resp. ] is the quotient projection from the free semigroup with involution on [resp. the free monoid with involution on ] and the free group on .
|Date of creation||2013-03-22 16:11:47|
|Last modified on||2013-03-22 16:11:47|
|Last modified by||Mazzu (14365)|