Lattice Theory

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9400511 Nation This research is concerned with the study of the equational theory of finite lattices and the structure of free lattices. These problems are related because questions about lattice varieties can often be interpreted as problems in free lattices. The first objective of this work is to determine which finitely generated equational classes of lattices have a term rewrite system. The second objective is to find an algorithm for determining all lattice varieties which cover a given finitely generated lattice variety. The principal investigator will also study which ordered sets can be embedded in free lattices, and which lattices are sublattices of free lattices. 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.