Authors: Jiménez, Jorge
Montes, Susana
Šešelja, Branimir
Tepavčević, Andreja 
Title: On lattice valued up-sets and down-sets
Journal: Fuzzy Sets and Systems
Volume: 161
Issue: 12
First page: 1699
Last page: 1710
Issue Date: 16-Jun-2010
Rank: M21a
ISSN: 0165-0114
DOI: 10.1016/j.fss.2009.11.012
Abstract: 
Isotone and anti-isotone mappings from a poset into a complete lattice are investigated as lattice-valued up-sets and down-sets, respectively. Cuts of these are shown to be analogue crisp sub-posets of the domain: up-set or semi-filters and down-sets or semi-ideals. The collection of all lattice-valued up-sets (down-sets) of a poset is a complete lattice under the order inherited from the lattice. Among other results, for a collection of crisp up-sets (down-sets) of a poset, necessary and sufficient conditions are given under which this collection consists of cuts of a lattice valued up-set (down-sets). A generalization in the sense of closed fuzzy sets with respect to fuzzy relations is also carried out.
Keywords: Closedness | L-valued down-sets | L-valued sets | L-valued up-sets
Publisher: Elsevier
Project: Serbian Ministry of Science and Environment, Grant no. 144011
Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant “Lattice methods and applications”

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