Geodesic
The equational theory of the weak Bruhat order on finite symmetric groups
Journal of the European Mathematical Society, Tome 20 (2018) no. 8, pp. 1959-2003.

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It is well known that the weak Bruhat order on the symmetric group on a finite number n of letters is a lattice, denoted by P(n) and often called the permutohedron on n letters, of which the Tamari lattice A(n) is a lattice retract. The equational theory (or word problem) of a class of lattices is the set of all lattice identities satisfied by all members of that class. Our main results imply, as particular cases, the following.
DOI : 10.4171/jems/804
Classification : 06-XX, 03-XX, 20-XX
Mots-clés : Lattice, identity, weak order, permutohedron, Cambrian lattice, Tamari lattice, monadic second-order logic, decidability, score, bounded homomorphic image, subdirectly irreducible, splitting lattice, splitting identity, polarized measure, sub-tensor product, box product, dismantlable lattice 
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     title = {The equational theory of the weak {Bruhat} order on finite symmetric groups},
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Luigi Santocanale; Friedrich Wehrung. The equational theory of the weak Bruhat order on finite symmetric groups. Journal of the European Mathematical Society, Tome 20 (2018) no. 8, pp. 1959-2003. doi : 10.4171/jems/804. https://geodesic-test.mathdoc.fr/articles/10.4171/jems/804/

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