Geodesic
On the Brauer Monoid of~$S_3$
Lobachevskii journal of mathematics, Tome 14 (2004), pp. 3-16.

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In $[{\rm HLS}]$, the authors showed that the Brauer monoid of a finite Galois group can be written as a disjoint union of smaller pieces (groups). Each group can be computed following Stimets by defining a chain complex and checking its exactness. However, this method is not so encouraging because of the difficulty of dealing with such computations even with small groups. Unfortunately, this is the only known method so far. This paper is to apply Stimets' method to some idempotent weak 2-cocycles defined on $S_3$. In particular, the idempotent 2-cocycles whose associated graphs have two generators. Some nice results appear in the theory of noncommutative polynomials. 
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A. Aljouiee. On the Brauer Monoid of~$S_3$. Lobachevskii journal of mathematics, Tome 14 (2004), pp. 3-16. http://geodesic.mathdoc.fr/item/LJM_2004_14_a0/

[A] A. Aljouiee, “On Weak Crossed Products, Frobenius Algebras and Weak Bruhat Ordering”, J. Algebra, 287:1 (2005), 88–102 | DOI | MR | Zbl

[H1] D. Haile, “On Crossed Product Algebras Arising from Weak Cocycles”, J. Algebra, 74 (1982), 270–279 | DOI | MR | Zbl

[H2] D. Haile, “The Brauer Monoid of a Field”, J. Algebra, 81 (1983), 521–539 | DOI | MR | Zbl

[HLS] D. Haile, R. Larson, M. Sweedler, “Almost Invertible Cohomology Theory and the Classification of Idempotent Cohomology Classes and Algebras by Partially Ordered Sets with a Galois Group Action”, Amer. J. Math., 105 (1983), 689–814 | DOI | MR | Zbl

[S1] R. Stimets, “Weak Galois Cohomology and Group Extension”, Communications in Algebra, 28:3 (2000), 1285–1308 | DOI | MR | Zbl

[S2] R. Stimets, “Relative Weak Cohomology and Extension”, J. Algebra (to appear)