Mathematical Sciences: Varieties of Lattices

本研究将集中于有限代数和有限格结构的三个问题。首先研究有限代数在满足非平凡Mal'cev条件的任意代数群中的同余结构。第二步是找到可以表示为置换等价关系格的有限长度格的描述。第三步是找出覆盖有限格所生成的格变体。格是离散的数学对象，其结构在一定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可以在许多数学环境中找到格。这也意味着问题很容易表述，很难解决，并且具有非常普遍的性质。Nation在过去是非常成功的，人们可以期待在格子上的结果，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.