Continuous Order Homomorphism and bi-continuous Mapping

The relation of LFtopology on set X to the classical topology is always important for us to understand LFtopology. In this paper, we construct a bitopological space from a LFtopologial space,and we show that for a continuous order homomorphism between two LFtopological spaces there is a correspondent bicontinuous mapping between the constructed bitopological spaces.Let X be a set,L be a fuzzy lattice,use L~x denote all the fuzzy subsets of X use π_x denote set X×{L\{ O, 1} }, P(πx) is power set of π_x. Now we first define mapping, a of π_x into itself as following: