Authors: Šešelja, Branimir
Tepavčević, Andreja 
Title: Representation of lattices by fuzzy sets
Journal: Information Sciences
Volume: 79
Issue: 3-4
First page: 171
Last page: 180
Issue Date: 1-Jan-1994
Rank: M23
ISSN: 0020-0255
DOI: 10.1016/0020-0255(94)90117-1
We prove that every lattice L of finite length can be represented by a fuzzy set on the collection X of meet-irreducible elements of L. A decomposition of this fuzzy set gives a family of isotone functions from X to 2 = ({0,1}, ≤), the lattice of which is isomorphic to L. More generally, conditions under which any collection of isotone functions from a finite set into 2 corresponds to a decomposition of a fuzzy set are given. As a consequence, the representation theorem for a finite distributive lattice by the lattice of all isotone functions is obtained. The collection of all lattices characterized by the same fuzzy set turns out to be a lattice with the above-mentioned distributive lattice as the greatest element.
Keywords: Fuzzy set decomposition | Isotone functions | Lattice representation | Meet-irreducible elements
Publisher: Elsevier

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