1

Let $S$ be a monoid generated by $\{x_1,x_2,x_3,x_4,x_5\}$ with the following relations: $x_i^2=0$ for all $i$, $x_ix_j=x_jx_i$ for all $|i-j| \geq 2$ and $x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}$ for all $i$. I want to apply the Bergman's Diamond lemma to find out a set of irreducible words. How do I do this ? What are all possible ambiguities and what is a possible set of irreducible words ?

user73789
  • 143
  • The monoid you consider is a Coxeter group of type $A_5$. So it is isomorphic to $S_6$, the symmetric group on $6$ letters (one can for example identify $x_i$ with the transposition that exchanges the $i$-th and $j$-th letter). – Ewan Delanoy May 06 '13 at 12:08
  • @ Ewan Delanoy, Here $x_i^2=0$ not 1. Anyway I just want to learn how the Diamond lemma works in this case. – user73789 May 07 '13 at 06:50
  • you mix up additive and multiplicative notation and this is rather confusing. Are you considering one or two operations on your base set $S$ ? If, as the words “monoid” and “semigroup” suggests in your post, there is only one operation, then the identity element will be denoted by $0$ or $1$ according to whether you write the operation additively or multiplicatively. But there will be no sense in distinguishing $0$ from $1$. – Ewan Delanoy May 07 '13 at 07:58
  • @Ewan: There is a lot of sense in distinguishing $0$ from $1$. In semigroups, $1$ is usually used to represent a multiplicative identity, while $0$ is used to represent a multiplicative zero, i.e. an element such that $x0 = 0x = 0$ for all $x\in S$. There is no mixing up of additive and multiplicative notation going on here. – Tara B May 10 '13 at 11:18
  • This is a special case of the more general question http://math.stackexchange.com/questions/369214/semigroup-presentation-and-diamond-lemma – J.-E. Pin Aug 28 '13 at 23:06
  • This is the nilCoxeter monoid. The irreducible words are exactly the same as the irreducible words for the corresponding Coxeter group because Tits's solution to the word problem works simultaneously for the Coxeter group, the 0-Hecke monoid and the nilCoxeter monoid. I am not sure the Diamond lemma is the right approach to take here. Look in Ken Brown's book on Buildings and adopt the Coxeter group solution to the nilCoxeter version. The key point is that any two words of minimal length representing an element are connected by just braid relations (which preserve length). – Benjamin Steinberg Sep 11 '13 at 02:35

0 Answers0