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