An interesting problem in universal algebra is the connection between the internal structure of an algebra and the identities which it satisfies. The study of varieties of algebras provides some insight into this problem. Here we are concerned mainly with lattice varieties, about which a wealth of information has been obtained in the last twenty years. We begin with some preliminary results from universal algebra and lattice theory. The next chapter presents some properties of the lattice of all lattice sub-varieties. Here we also discuss the important notion of a splitting pair of varieties and give several characterisations of the associated splitting lattice. The more detailed study of lattice varieties splits naturally into the study of modular lattice varieties and non-modular lattice varieties, dealt with in the second and third chapter respectively. Among the results discussed there are Freese's theorem that the variety of all modular lattices is not generated by its finite members, and several results concerning the question which varieties cover a given variety. The fourth chapter contains a proof of Baker's finite basis theorem and some results about the join of finitely based lattice varieties. Included in the last chapter is a characterisation of the amalgamation classes of certain congruence distributive varieties and the result that there are only three lattice varieties which have the amalgamation property.
Reference:
Jipsen, P. 1988. Varieties of lattices. University of Cape Town.
Jipsen, P. (1988). Varieties of lattices. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/15851
Jipsen, Peter. "Varieties of lattices." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988. http://hdl.handle.net/11427/15851
Jipsen P. Varieties of lattices. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/15851