The equational theory of the weak Bruhat order on finite symmetric groups
The weak Bruhat order on the symmetric group \(S_n\) is a lattice \(\mathsf{P}(n)\) called the permutohedron on \(n\) letters. The Tamari lattice \(\mathsf{A}(n)\) is a lattice retract of \(\mathsf{P}(n)\). Given a poset \(E\), there is an `extended permutahedron' lattice defined from \(E\) denoted \(\mathsf{R}(E)\). The main results of this important paper are: A. The class of all \(\mathsf{P}(n)\) has a decidable equational theory. B. The class of all \(\mathsf{A}(n)\) has a decidable equational theory. C. There is a lattice identity that holds in all \(\mathsf{P}(n)\) but fails in some \(3,338\)-element lattice. D. Any finite meet-semidistributive lattice embeds in \(\mathsf{R}(E)\) for some countable poset \(E\). (The poset \(E\) for this embedding can always be taken to be the directed union of finite dismantlable lattices.) In particular, the class of all \(\mathsf{R}(E)\) generates the variety of all lattices.
scientific article; zbMATH DE number 6925183