Grid Point Consensus Structure With Point Measure Theory And Its Applications

In this paper, our main purpose is to build a new theory about uniformity and metric on completely distributive lattices (or fuzzy lattices) arid present. its applications in Domain area of theoretical computer science. This paper is consist of four chapt.ers. In first chapter, our main purpose is t.o build a new theory about uniformity on completely distribu- tive lattices (or fuzzy lattices) such that it can directly reflect the characteristics of pointwise topology on lattices, i.e. it can directly reflect. the relation between a point. (or molecule) and its Q-neighborhood or R-neighborhood. Based on the research of properties of remote-neighborhood mappings and the charact.erization of crisp uniforrriity, we introduce a new definition of quasi-uniformity on completely distributive lattices which is called pointwise quasi-uniformity. It. is proved that each topological molecular lattice is pointwise quasi-u niformizable. Meanwhile we introduce the concept of pointwise quasi-uniform continuous general- ized order-homomorphism. We have the category PQIJML in which each object. is a topological molecular lattice and each morphism is point.wise quasi-uniform continuous generalized order-homomorphism. In this category, the product. of topological molecular lattices is closed. Based on these work mentioned above, by means of order-reversing involution, we introduce the concept. of pointwise uniformity on fuzzy lattices. A pointwise uniformity can be charact.erized in terms of remote-neighborhood mappings. To characterize a pointwise uniformity, we define pointwise complete regularity. A topology on fuzzy lattices can be induced by a point.wise uniformity if and only if it. is point- ~vse completely regular. The L-fuzzy unit interval and the L-fuzzy real line are pointwise uniformizable. Moreover the relation between pointwise uniformities and Hutton uniformities is given and pointwise complete regularit.y is characterized by the L-fuzzy unit interval. It is proved that An L-fuzzy topological space is pointwise completely regular sub-To space if and only if it is homeomorphic with a subspace of the product space of copies of the L-fuzzv unit intervals. In second chapter, our maui purpose is t.o build a new theory about metric on completely distributive lattices (or fuzzy lattices) such that it can directly reflect the characteristics of pointwise topology on lattices, i.e. it. can directly reflect the relation bet.ween a point. (or molecule) and it.s Q-neighborhood or R-neighborhood. First the concept of point.wise p.q. metric is int.roduced on completely distributive !at.tices and it.s characterization is given by means of remote-neighborhood mappings. The cotopology of a pointwise p.q. metric is C1. It is pointed that we have a category PPQMML, whose objects are pointwise p.q. metric molecular lattices and whose morphism are pointwise quasi-uniform continuous generalized order-homomorphism, which is called the cat.egory of pointwise p.q. metric molecular lattices. It is a full subcat.egory of PQUML A pointwise qua.si-uniformit.y can be induced by a point.wise p.q. metric if and only if it. has a count.al)le basis. Further we give a simple proof that a C11 topological molecular lattice is point.wise p.q. metrizable. It is also proved that t.he product. of a countable family of topological molecular lat.tices is still a topological molecular lattice. Moreover, the concept. of pointwise p. metric and pointwise metric are also introduced on fuzzy lattices. The L-fuzzy unit interval and the L-fuzzv real line are point.wise p. mnet.rizable.