Representation of lattices via set-colored posets

This paper proposes a representation theory for any finite lattice via set-colored posets, in the spirit of Birkhoff for distributive lattices. The notion of colored posets was introduced in Nourine (2000) [34] and the generalization to set-colored posets was given in Nourine (2000) [35]. In this pa...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2018-11, Vol.249, p.64-73
Main Authors: Habib, Michel, Nourine, Lhouari
Format: Article
Language:English
Subjects:
Antimatroid
Applied mathematics
Closure system
Coloring
Computer Science
Discrete element method
Discrete Mathematics
Lattice
Lattice theory
Lattices
Representation theory
Representations
Set theory
Set-colored poset
Upper locally distributive lattice
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Summary:This paper proposes a representation theory for any finite lattice via set-colored posets, in the spirit of Birkhoff for distributive lattices. The notion of colored posets was introduced in Nourine (2000) [34] and the generalization to set-colored posets was given in Nourine (2000) [35]. In this paper, we give a characterization of set-colored posets for general lattices, and show that set-colored posets capture the order induced by join-irreducible elements of a lattice as Birkhoff’s representation does for distributive lattices. We also give a classification for some lattices according to the coloring property of their set-colored representation including upper locally distributive, upper locally distributive, meet-extremal and semidistributive lattices.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2018.03.068