The Partial Order In Interval On Lattices

In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra.In a lattice, two intervals [a, b] and [a’, b’] are called projective which can be written as a finite sequence [a,b], [a1,b1], [a2,b2],…, [an,bn] and [a’, b’] in which any two successive intervals are transpositive. This idea goes back to Birkhoff. A natural generalization is to consider not only transpositive but also inclusion relation. The concept of weak projectivity is due to Dilworth. He implemented a detailed discussion and illustration on the properties of the weak projectivity on lattices, and gave a necessary conditions for one interval to be weakly projective into another on modular lattices. According to that conclusion, we define an especial partial order which is based on the given interval and introduce the concept of tail on complete lattices. In that case, some useful conclusions are drawn from the analysis.In Chapter 3 and Chapter 4, we consider the weak projectivity on power structures on lattices and quantum logic respectively.Power structures epitomize the attempt to lift some existing structure from elements of a set to subsets of that set. Considering the power structures on lattices, Hu first stepped toward the upgrade problem for lattices, established the concept of upgrade lattice and showed the condition of being an upgrade lattice. And then he discussed some properties of the upgrade lattice and ideals, homomorphisms, upgrade lattice groups on the upgrade lattice. This goal of the chapter can be stated very simply:to construct some isomorphic upgrade lattices on some special lattices by taking the advantage of weak projectivity. We first show that lattices, con-structed by lifted order relation, are profoundly different from which upgraded from join and meet operations. And then our emphasis is put on inner-disjoint-upgrade-lattices, distributive lattice and relatively complemented lattices, in the case, we start to study equivalence class on latttices. Finally a constructional method of isomorphic mapping on lattice is released in the sense of upgrade lattices.For quantum logic the most natural candidate for a model is the orthomodular lattice. In fact, the orthomodular lattice as a lattice with some special properties, is the weak form of Boolean algebra from the aspect of distributivity. For the algebraist quantum logic is just an exotic name for well-defined mathematical structure, which is part of a well-established branch of mathematics (lattice theory). From the perspective of weak projectivity, we study the relationships between the orthomodular lattice and Boolean algebra and show the compat-ibility of the tail and the meet and the properties of tails.