Timeline for Derivable relations in a monoid
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2021 at 9:26 | comment | added | diddy | Anyway, this is a neat trick. Thank you for explaining it. | |
| Mar 14, 2021 at 9:25 | comment | added | diddy | And you were right, it is not the reversed word. I was implicitely reverseing the $ y $ word. | |
| Mar 14, 2021 at 9:18 | comment | added | diddy | @BenjaminSteinberg Ok, I am not familiar with the terminology. If one says normal form, then it means the word in the free monoid, not the presentation in the monoid with relations. Then it makes sense. I was missing that the point is to interpret the monoid as the operator monoid. Instead I was focusing on the set of normal forms. | |
| Mar 14, 2021 at 3:41 | comment | added | Benjamin Steinberg | If you are acting on the left then i think you should get the normal form from the empty word not the reversed. That means the normal form Is unique because different normal forms act differently on the empty word | |
| Mar 13, 2021 at 17:19 | comment | added | diddy | @BenjaminSteinberg then I don't get how this operation provides the uniqueness. I get that the operators satisfy the defining relations and that a $ y $ word applied on the empty word yields the (I think reversed) $ x_{rev} $-word. How to continue from here? | |
| Mar 13, 2021 at 16:40 | answer | added | diddy | timeline score: 1 | |
| Mar 13, 2021 at 15:57 | comment | added | Benjamin Steinberg | I think it's just a standard name for the trick where you solve the word problem for a monoid or group by having our act on the set of normal forms. I learned that terminology in grad school | |
| Mar 13, 2021 at 15:08 | comment | added | diddy | @BenjaminSteinberg thank you again. Your first suggestion seems to work. I will write an answer using this idea later (if didn't make a mistake). For your second suggestion: I get the operation. But, I couldn't find a reference in which the van der Waerden trick is presented in a (at least for me) useful way. Do you have such a reference? I am curious. | |
| Mar 13, 2021 at 13:21 | comment | added | Benjamin Steinberg | You can build operators $y_i$ and $\hat{y_i}$ that act on your normal forms by $y_i$ places $x_i$ at the left end and $\hat{y_i}$ removes the left most occurrence of $x_i$ if there is any and if not it inserts $\hat{x_i}$ as the leftmost hatted letter. You can check these satisfy your defining relations and a y-word sends the empty normal form to the x-word normal form of the y-word | |
| Mar 13, 2021 at 13:16 | comment | added | Benjamin Steinberg | This is a complete rewriting system I believe and your normal forms are the irreducible forms so they are distinct. You could also build an action on those normal forms to use the van der Waerden trick | |
| Mar 13, 2021 at 12:10 | history | asked | diddy | CC BY-SA 4.0 |