Mathematical Sciences: Universal Algebra and Lattice Theory

本研究涉及自由有限表示格的结构理论。主要研究人员将使用自由格中的覆盖理论和某些无限技巧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，而后者都是无限的。有限表示的晶格何时是弱原子，何时是有限宽度的问题也将被研究。将寻求一种有效的算法来判定有限表示格是否有限。我们还将研究这样一个猜想，即一个无限宽的有限表示格一定包含一个子格，它与三个生成元上的自由格同构。格是离散的数学对象，其结构在某种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格子可以在许多数学环境中找到。这也意味着问题很容易陈述，很难解决，而且具有非常普遍的性质。这项工作的成果将对数学、计算机科学、社会科学和企业管理的许多领域产生影响。

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.