Types Of Sequence Elements Of The Algebra Decomposition Theory

The decompositions of an element in distributive lattices, pseudocomple-mented modular lattices and effect algebras are discussed. The factor theo-rems play an important role in number theorem, polynomial algebra and ring theory. But in many ordered algebras, some form of decomposition of an ele-ment may not exist, or be not unique. The research on theses special elements in algebra systems can help us to better understand their structures. This thesis mainly focuses on the elements with atomless representation in dis-tributive or pseudocomplemented modular lattices, and the torsion elements which originate from the element that can be decomposed into the sum of atoms in more than one "basic" ways in effect algebras. The specific work we have done is as follows:(1) The calculations of the spectrum of the direct product and ordinal sum of lattices are given.(2) By adding the elements with atomless representation into the library of factors, we propose the minimal representation of an element in lattices satisfying the descending chain condition. Some properties of the minimal representation are discussed.(3) We shown that the minimal representation of a given element in dis-tributive lattices is unique. Using this result, we characterize the sequences that can be represented by distributive lattices and discuss to what extent does the sequence represented by a distributive lattice or a boole lattice char-acterize the structure of it. These results settle a problem on the spectrums of distributive lattices raised by G. Gratzer, D. S. Gunderson and R. W. Quackenbush, and make some progresses on another one.(4) We shown that the minimal representation of an element in pseudo- complemented modular lattices is unique and characterize the sequences that can be represented by pseudocomplemented modular lattices. These results settle a problem on the spectrums of pseudocomplemented modular lattices raised by G. Gratzer, D. S. Gunderson and R. W. Quackenbush.(5) In effect algebras, the elements that can decompose into the sum of the atoms in more than one "basic" ways are defined as torsion elements, whose physical background is quantum entanglement. We show that an or-thocomplete effect algebra is an MV-effect algebra if and only if it has no torsion element, an orthocomplete effect algebra is homogeneous if and only if it has no unsharp torsion element. We generalize the basic decomposition of an element in lattice effect algebras into orthocomplete homogeneous effect algebras, prove that every element of an orthocomplete homogeneous atomic effect algebra has a unique basic decomposition into a sum of a sharp element and unsharp multiples of atoms. Further, we characterize homogeneity by the set of all sharp elements and the atom decompositions of the unit 1 in orthocomplete atomic effect algebras, respectively.