Tamari lattices and the symmetric Thompson monoid. (English) Zbl 1284.06004

Dehornoy, Patrick

Müller-Hoissen, Folkert (ed.) et al., Associahedra, Tamari lattices and related structures. Tamari memorial Festschrift. Basel: Birkhäuser (ISBN 978-3-0348-0404-2/hbk; 978-3-0348-0405-9/ebook). Progress in Mathematics 299, 211-250 (2012).

Summary: We investigate the connection between Tamari lattices and the Thompson group \( F \), summarized in the fact that \( F \) is a group of fractions for a certain monoid \(F^+_{\mathrm{sym}} \) whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice meet and join are the counterparts of the least common multiple and greatest common divisor operations in \(F^+_{\mathrm{sym}}\). As an application, we show that, for every \( n \), there exists a length \( \ell \) chain in the \( n \)th Tamari lattice whose endpoints are at distance at most \(12 \ell/n \).
For the entire collection see [Zbl 1253.00013].

MSC:

06A07	Combinatorics of partially ordered sets
05E10	Combinatorial aspects of representation theory
20F65	Geometric group theory

Keywords:

Tamari lattices; symmetric Thompson monoid; Thompson group