The equational theory of the weak Bruhat order on finite symmetric groups
type: Item
instance of: scholarly article
title: The equational theory of the weak Bruhat order on finite symmetric groups
zbMATH Open document ID: 1450.06001
DOI: 10.4171/JEMS/804
author: Luigi Santocanale
author: Friedrich Wehrung
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2018-08-24T00:00:00Z
review text: 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.
reviewed by: Keith A. Kearnes
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 03C85
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06B25
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 6925183
zbMATH Keywords: lattice
zbMATH Keywords: identity
zbMATH Keywords: weak order
zbMATH Keywords: permutohedron
zbMATH Keywords: Cambrian lattice
zbMATH Keywords: Tamari lattice
zbMATH Keywords: monadic second-order logic
zbMATH Keywords: decidability
zbMATH Keywords: score
zbMATH Keywords: bounded homomorphic image
zbMATH Keywords: subdirectly irreducible
zbMATH Keywords: splitting lattice
zbMATH Keywords: splitting identity
zbMATH Keywords: polarized measure
zbMATH Keywords: sub-tensor product
zbMATH Keywords: box product
zbMATH Keywords: dismantlable lattice
MaRDI profile type: Publication
Wikidata QID: Q129718461
description: scientific article; zbMATH DE number 6925183