Mathematical Sciences: Lattice Theory

数学科学：格论

• 批准号: 9500752
• 负责人: Ralph Freese
• 金额: $ 10.93万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500752&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Lattice; Theory

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.

该奖项支持对有限格的结构的研究以及有限格的算法问题的研究。 在新提出的格中，将寻求基本结构和表示问题的有效解决方案。 这类问题包括找到一个多项式时间算法来决定一个非线性表示格的元素是否有下覆盖，非线性表示格是否有限，是否是弱原子的，以及是否有有限的宽度。 一个猜想的证明，即一个无限宽的非对称格必须包含一个自由格将寻求。 此外，更快的算法识别和有限格的建设将进行调查。 利用关于自由格、双表示格和射影格的定理，我们将寻求求解自由格中方程组的有效方法。 这些方法应导致有效的算法，用于确定这种系统的统一类型。 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可以在许多数学环境中找到。 这也意味着问题容易陈述，难以解决，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，社会科学和企业管理的许多领域。