Premium
An Extension of Dilworth's Theorem
Author(s) -
Sims Julie A.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-16.3.393
Subject(s) - mathematics , lattice (music) , isomorphism (crystallography) , embedding , function space , discrete mathematics , extension (predicate logic) , pure mathematics , combinatorics , crystal structure , computer science , physics , chemistry , artificial intelligence , acoustics , programming language , crystallography
We show that any lattice of finite length is isomorphic to the lattice of fully dependent flats of a rank‐finite independence space, which is finite if the given lattice is finite. If a suitable submodular function is given on the lattice, the independence space may be so chosen that this submodular function corresponds to the rank function of the independence space under the constructed isomorphism. The isomorphism also has the property that it constitutes an embedding of the given lattice in the (geometric) lattice of flats of the independence space.