Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius

Andrea Brini

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-A, Issue: 1, page 143-172
  • ISSN: 0392-4041

Abstract

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Si introduce la nozione combinatoria di connessione di Galois tra insiemi parzialmente ordinati e se ne descrivono i principali risultati di caratterizzazione; questi risultati aprono la strada alla comprensione del profondo legame che sussiste tra la nozione connessione di Galois ed il Criterio di Omotopia di Quillen. Si introduce quindi la nozione di funzione di Möbius di un reticolo finito L e se ne discute brevemente, anche tramite un esempio significativo, la cruciale importanza nell’ambito della Combinatoria Enumerativa e della Probabilità Discreta. Dopo aver riconosciuto che i valori della funzioni di Möbius possono essere interpretati come «Caratteristiche di Eulero» di opportuni complessi, a titolo di esempio e di applicazione di metodi topologici alla combinatoria degli insiemi parzialmente ordinati, si presentano e si dimostrano le versioni topologiche di due classici Teoremi: il «Teorema del Cross-Cut» ed il «Teorema di annullamento per reticoli non fortemente complementati».

How to cite

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Brini, Andrea. "Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius." Bollettino dell'Unione Matematica Italiana 7-A.1 (2004): 143-172. <http://eudml.org/doc/289415>.

@article{Brini2004,
abstract = {The notion of Galois Connections between partially ordered sets is introduced, together with a presentation of some of its main characterizations. This leads to a true understanding of the deep connection that links Galois Connections to Quillen’s Homotopy Type Equivalence Theorem. Furthermore, the notion of Möbius functions of finite lattices is discussed, in order to show its crucial role in Enumerative Combinatorics over Finite Posets and Discrete Probability Theory. Since the values of the Möbius function of a finite lattice may be regarded as reduced Euler Characteristic of suitable topological spaces, a wide variety of combinatorial results can be proved by topological methods. We exploit this point of view by providing elementary proofs of two classical theorems: the «Cross-Cut Theorem» of Rota and the «Vanishing Theorem for not-strongly complemented lattices» of Crapo.},
author = {Brini, Andrea},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {ita},
month = {4},
number = {1},
pages = {143-172},
publisher = {Unione Mastematica Italiana},
title = {Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius},
url = {http://eudml.org/doc/289415},
volume = {7-A},
year = {2004},
}

TY - JOUR
AU - Brini, Andrea
TI - Combinatoria e Topologia. Teorema di Quillen e funzioni di Möbius
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/4//
PB - Unione Mastematica Italiana
VL - 7-A
IS - 1
SP - 143
EP - 172
AB - The notion of Galois Connections between partially ordered sets is introduced, together with a presentation of some of its main characterizations. This leads to a true understanding of the deep connection that links Galois Connections to Quillen’s Homotopy Type Equivalence Theorem. Furthermore, the notion of Möbius functions of finite lattices is discussed, in order to show its crucial role in Enumerative Combinatorics over Finite Posets and Discrete Probability Theory. Since the values of the Möbius function of a finite lattice may be regarded as reduced Euler Characteristic of suitable topological spaces, a wide variety of combinatorial results can be proved by topological methods. We exploit this point of view by providing elementary proofs of two classical theorems: the «Cross-Cut Theorem» of Rota and the «Vanishing Theorem for not-strongly complemented lattices» of Crapo.
LA - ita
UR - http://eudml.org/doc/289415
ER -

References

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