On Tarski’s Type Theorems for L-Fuzzy Isotone and L-Fuzzy Relatively Isotone Maps on L-Complete Propelattices

Recently a new type of very general relational
structures, the so called (L-)complete propelattices, was introduced.
These significantly generalize complete lattices and completely lattice
L-ordered sets, because they do not assume the technically very
strong property of transitivity. For these structures also the main part
of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone
maps, i.e., the part which concerns the existence of fixed points and
the structure of their set. In this paper, fundamental properties of
(L-)complete propelattices are recalled and the so called L-fuzzy
relatively isotone maps are introduced. For these maps it is proved
that they also have fixed points in L-complete propelattices, even if
their set does not have to be of an awaited analogous structure of
a complete propelattice.

Authors:

• František Včelař
• Zuzana Pátíková

Keywords:

• Fixed point
• L-complete propelattice
• L-fuzzy (relatively) isotone map
• residuated lattice
• transitivity.

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