Some results on $L$-complete lattices

TL;DR

This paper explores properties of $L$-fuzzy complete lattices and fuzzy dcpos, establishing theorems on monotone maps, fixpoints, and the structure of fuzzy hom-sets, advancing the theory of fuzzy order structures.

Contribution

It introduces new theorems on monotone maps, fixpoints, and fuzzy hom-sets in $L$-fuzzy complete lattices and fuzzy dcpos, expanding the understanding of fuzzy order theory.

Findings

Monotone maps on $L$-fuzzy complete lattices have least fixpoints.

The structure of $Hom(P,P)$ as a fuzzy $dcpo$ is established.

Fuzzy versions of key order-theoretic rules are formulated.

Abstract

The paper deals with special types of $L$-ordered set, $L$-fuzzy complete lattices, and fuzzy directed complete posets (fuzzy $dcpo$s). First, a theorem for constructing monotone maps is proved, a characterization for monotone maps on an $L$-fuzzy complete lattice is obtained, and it is proved that if $f$ is a monotone map on an $L$-fuzzy complete lattice $(P;e)$, then $\sqcap S_f$ is the least fixpoint of $f$. A relation between $L$-fuzzy complete lattices and fixpoints is found and fuzzy versions of monotonicity, rolling, fusion and exchange rules on $L$-complete lattices are stated. Finally, we investigate $Hom(P,P)$, where $(P;e)$ is a fuzzy $dcpo$, and we show that $Hom(P,P)$ is a fuzzy $dcpo$, the map $\gamma\mapsto \bigwedge_{x\in P}e(x,\gamma(x))$ is a fuzzy directed subset of $Hom(P,P)$, and we investigate its join.