Mathematical Sciences: Universal Algebra and Lattice Theory

点击翻译按钮获取中文摘要

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.