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**Cayley lattices of finite Coxeter groups are bounded.**
*(English)*
Zbl 1097.06001

Given a poset \(P\) and an interval \(I\) in \(P\), an interval doubling \(P(I)\) replaces \(I\times\{0\leq1\}\) maintaining the order \(x< I\) by \(x< I\times\{0\}\) and \(x> I\) by \(x> I\times\{1\}\). If \(P\) can be obtained from \(\{0\leq 1\}\) by a finite sequence of interval doublings, then it is bounded. In this useful paper a class of lattices HH is introduced, probably of importance in its own right, whose elements are shown to be bounded. Coxeter lattices are obtained from the transitive closures of the Cayley graphs of finite Coxeter groups via the weak order for general Coxeter groups. They are shown to be members of HH, which makes HH an interesting class of lattices ipso facto and bounded. The relationship between construction and deconstruction as in HH can be observed on the Coxeter lattices with meaningful interpretations in terms of reflections and parabolic subgroups whose lattices may each be reached from the over-arching lattice by a series of interval contractions as shown. Due to the intimate connections between Coxeter groups and geometry, the notions and results introduced and obtained here also yield new interpretations via these connections for even greater interest as noted by the authors.

Reviewer: Joseph Neggers (Tuscaloosa)

### MSC:

06A07 | Combinatorics of partially ordered sets |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

06B05 | Structure theory of lattices |

### Keywords:

Bounded lattice; Coxeter group; Doubling operation; Reflection; Semidistributivity; Zonotope; Coxeter lattice
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\textit{N. Caspard} et al., Adv. Appl. Math. 33, No. 1, 71--94 (2004; Zbl 1097.06001)

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### References:

[1] | Barbut, M; Monjardet, B, Ordre et classification, (1970), Hachette Paris, France, vols. 1 and 2 · Zbl 0267.06001 |

[2] | Birkhoff, G, Lattice theory, (1940), Amer. Math. Soc Providence, RI · Zbl 0126.03801 |

[3] | Björner, A, Orderings of Coxeter groups, (), 175-195 |

[4] | Björner, A; Wachs, M, Generalized quotients in Coxeter groups, Trans. amer. math. soc., 308, 1, (1988) · Zbl 0659.05007 |

[5] | Bourbaki, N, Groupes et algèbres de Lie, chapitres 4, 5, 6, Éléments de mathématiques, vol. 34, (1968), Hermann Paris |

[6] | Caspard, N, The lattice of permutations is bounded, Internat. J. algebra comput., 10, 4, 481-489, (2000) · Zbl 1008.06004 |

[7] | N. Caspard, C. Le Conte de Poly-Barbut, Tamari lattices are bounded: a new proof, Preprint, 2003 |

[8] | N. Caspard, H. Crapo, C. Le Conte de Poly-Barbut, Lines arrangements of reflections in Coxeter groups, Preprint, 2003 |

[9] | Coxeter, H.S.M, The classification of zonohedra by means of projective diagrams, J. math. pures appl., 41, 137-156, (1962) · Zbl 0123.13701 |

[10] | Davey, B.A; Priestley, H.A, Introduction to lattices and order, (1992), Cambridge University Press · Zbl 0701.06001 |

[11] | Day, A, A simple solution to the word problem for lattices, Canad. math. bull., 13, 253-254, (1970) · Zbl 0206.29702 |

[12] | Day, A, Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices, Canad. J. math., 31, 69-78, (1979) · Zbl 0432.06007 |

[13] | Day, A; Nation, J.B; Tschantz, S, Doubling convex sets in lattices and a generalized semidistributivity condition, Order, 6, 175-180, (1989) · Zbl 0695.06005 |

[14] | Day, A, Congruence normality: the characterization of the doubling class of convex sets, Algebra universalis, 31, 397-406, (1994) · Zbl 0804.06006 |

[15] | Geyer, W, The generalized doubling construction and formal concept analysis, Algebra universalis, 32, 341-367, (1994) · Zbl 0829.06007 |

[16] | Guilbaud, G; Rosenstiehl, P, Analyse algébrique d’un scrutin, Math. sci. hum., 4, 9-33, (1963) |

[17] | Humphreys, J.E, Reflection groups and Coxeter groups, (1990), Cambridge University Press Cambridge · Zbl 0725.20028 |

[18] | Le Conte de Poly-Barbut, C, Sur LES treillis de Coxeter finis, Math. inf. sci. hum., 125, 45-57, (1994) · Zbl 0802.06016 |

[19] | Le Conte de Poly-Barbut, C, Treillis de Cayley des groupes de Coxeter finis. constructions par récurrence et décompositions sur des quotients, Math. inf. sci. hum., 140, 11-33, (1997) · Zbl 0980.20029 |

[20] | McKenzie, R, Equational bases and non-modular lattice varieties, Trans. amer. math. soc., 174, 1-43, (1972) |

[21] | Wille, R, Subdirect decomposition of concept lattices, Algebra universalis, 17, 275-287, (1983) · Zbl 0539.06006 |

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