Mathematical Sciences: Universal Algebra and Lattice Theory

本文研究的是自由有限表示晶格的结构理论。主要研究者将使用自由格中的覆盖理论和某些无穷技术来确定自由格中是否存在元素bca，使得区间a/c是a/c和c/b的并并且后者都是无限的。有限呈现的晶格何时是弱原子的，以及何时是有限宽度的问题也将被研究。我们将寻求一种有效的算法来判定一个有限表示的格是否是有限的。本文还研究了一个无限宽度的有限呈现晶格必须包含一个与三个发生器上的自由晶格同构的子晶格的猜想。格是离散的数学对象，其结构在一定程度上反映了集合论或逻辑的结构。这种非常弱的结构意味着可以在许多数学环境中找到格。这也意味着问题很容易表述，很难解决，并且具有非常普遍的性质。这项工作的结果将影响数学、计算机科学、社会科学和商业管理的许多领域。

This research is concerned with the structure theory of free and finitely presented lattices. The principal investigator will use the theory of covers in free lattices and certain infinitary techniques to determine if there are elements bca in a free lattice such that the interval a/c is the union of a/c and c/b and both the latter are infinite. The problem of when a finitely presented lattice is weakly atomic, and when it is of finite width will also be investigated. An efficient algorithm will be sought for deciding if a finitely presented lattice is finite. The conjecture that a finitely presented lattice of infinite width must contain a sublattice isomorphic to the free lattice on three generators will also be investigated. Lattices are discrete mathematical objects with a structure reflecting to some extent the structure of set theory or logic. This very weak structure means that lattices can be found in many mathematical environments. It also means that problems are easy to state, hard to solve and of a very general nature. The results of this work will impact many areas of mathematics, computer science, the social sciences and business management.