Mathematical Sciences: Lattice Theory

该奖项支持对有限表示格结构的研究以及对有限格的算法问题的研究。在有限表示格中，将寻求基本结构和表示问题的有效解决方案。这些问题包括找到一个多项式时间算法来确定有限表示格的元素是否具有下覆盖，有限表示格是否有限，它是否是弱原子，以及它是否具有有限宽度。我们将寻求一个猜想的证明，即无限宽的有限表示格必包含一个自由格子。此外，还将研究识别和构造有限格的快速算法。利用自由格、有限表示格和射影格上的定理，我们将寻求在自由格中求解方程组的有效方法。这些方法应该会导致有效的算法来确定这样一个系统的统一类型。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 award supports research on the structure of finitely presented lattices as well as the study of algorithmic problems of finite lattices. In finitely presented lattices, efficient solutions to basic structural and representation problems will be sought. Such problems include finding a polynomial time algorithm to decide if an element of a finitely presented lattice has a lower cover, if a finitely presented lattice is finite, if it is weakly atomic, and if it has finite width. A proof of the conjecture that a finitely presented lattice of infinite width must contain a free lattice will be sought. Furthermore, faster algorithms for the recognition and construction of finite lattices will be investigated. Using theorems on free lattices, finitely presented lattices, and projective lattices, efficient methods for solving systems of equations in free lattices will be sought. These methods should lead to efficient algorithms for determining the unification type of such a system. 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.