On fuzzification of Tarski's fixed point theorem without transitivity

The aim of this paper is to present a fuzzification of Tarski's fixed point theorem without the assumption of transitivity. For this purpose a new structure - the so called L-complete propelattice, which generalizes complete lattices and completely lattice L-ordered sets, is introduced. Our results show that for L-fuzzy isotone maps on L-complete propelattices a variant of Tarski's fixed point theorem holds. Especially, the set of fixed points is nonempty and of a certain structure.