On lattice valued up-sets and down-sets

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.