Gröbner-Shirshov Bases for Braid Groups in Adyan-Thurston Generators

Abstract

In this paper, we give a Gröbner-Shirshov basis of the braid group B n+1 in Adyan-Thurston generators. We also deal with the braid group of type B n . As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup B + n+1 is the subsemigroup in B n+1 .

Thurston's Operations on the Braid Groups

“…Remark 3.1. A Gröbner -Shirshov basis for the braid groups has been already found before (see [5]). This result is based on the concept of Bokut' -Shiao's normal form for permutations [3].…”

“…Step 3. Let us find the intersection aRa ∩ ¬R b = {(1, 3), (2, 3), (2,4), (2,5), (2, 7), (2, 8), (6, 7), (6, 8)}, we see that this set satisfies to condition ii) of Lemma 1.2.…”

“…3 (the left side). We have (1,3), (1,4), (1,5), (1,6), (2,4), (2,6), (2, 5), (4, 5)}, (1,4), (1,5), (1,6), (2,4), (2,6), (3,4), (3,5), (3,6), (5,6)}, we see that (4, 5), (5, 6) ∈ R b ∪ aRa but (4, 6) / ∈ R b ∪ aRa, i.e., this set does not satisfy to the condition i) of Lemma 1.2.…”

“…Step 4. Let Rx = aRa ∧¬R b = {(1, 3), (2, 3), (2,4), (2,5), (2, 7), (2, 8), (6, 7), (6, 8)}, we have for the permutation x the following system of inequalities…”

Gröbner-Shirshov basis for the singular part of the Brauer semigroup

“…Furthermore, in [16] and [17], Gröbner-Shirshov bases for Schreier extensions of groups and for the Chinese monoid were defined separately. The reader is referred to [1,5,6,8,19,21,22] for some other recent papers about Gröbner-Shirshov bases.…”