Mathematical Sciences: Varieties of Lattices

本文主要研究有限代数和有限格结构的三个问题。第一个是研究满足非平凡Mal‘cev条件的任意代数的有限代数的同余结构。二是找到了有限长格的一种描述，它可以表示为置换等价关系格。三是找出复盖由有限格生成的簇的格簇。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。国家在过去一直非常成功，人们可以期待格子方面的结果，这将影响数学的许多领域。

This research will focus on three problems concerning the structure of finite algebras and finite lattices. The first is to study the congruence structure of finite algebras in any variety of algebras satisfying a nontrivial Mal'cev condition. The second is to find a description of those finite length lattices which can be represented as a lattice of permuting equivalence relations. The third is to find the lattice varieties covering the variety generated by a finite lattice. 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. Nation has been very successful in the past and one can expect results on lattices which will impact many areas of mathematics.