Epimorphisms and dominions in varieties of lattices

This work is a study of dominions in varieties of lattices. Given any sublattice L′ of a lattice L, the dominion D is the part of L that is controlled by L′, in the sense that for every lattice-homomorphism f with domain L, f|_D is determined by f|_L′. Dominions are trivial in the variety of all lattices (i.e., D is always equal to L′), but there are nontrivial dominions in many subvarieties.; Chapter 1 is introductory material. It contains definitions of most of the terms that are used frequently in this work. It also contains some basic results about dominions, and some basic results from lattice theory and universal algebra that are needed later.; In Chapter 2, dominions in the variety of distributive lattices are characterized. It is proved that all of these dominions become trivial in any larger variety of lattices.; Chapters 3, 4, and 5 are concerned with three different generalizations of distributivity. In Chapter 3, dominions are studied in the varieties generated by lattices of length 2. In these varieties, dominions of sublattices of FM(3) are completely determined. It is shown that all dominions in Var( M_ω) become trivial in the variety of modular lattices.; Chapter 4 shows that for any n, the variety of n-distributive lattices has nonsurjective epimorphisms. Nonsurjective epimorphisms are also produced in another related sequence of varieties.; Chapter 5 examines the variety of almost distributive lattices and its subvarieties. All of these subvarieties have nonsurjective epimorphisms. Attention is focused on the variety generated by a finite subdirectly irreducible almost distributive lattice.; Chapter 6 describes a way of finding, in a direct product L × L′ of subdirectly irreducible lattices, certain proper sublattices which are epimorphically included. In particular, there are nonsurjective epimorphisms in any variety with a finite bound on the length, or on the width, of its subdirectly irreducible members.; Chapter 7 consists of five unrelated sections. Section 7.1 examines when a sublattice must be cofinal in its dominion. In section 7.2 it is shown that dominions of distributive sublattices in larger varieties of lattices generate the variety of all lattices. Section 7.3 gives an example of a finite sublattice with an infinite dominion. Section 7.4 contains a result on dominions in products of varieties. Section 7.5 shows that dominions respect direct products.; Chapter 8 departs from pure lattice theory to examine orthomodular lattices, which are a kind of lattice with additional operations. Various results about dominions in varieties of orthomodular lattices are proved. Some of these results are analogues of results about lattices from earlier chapters.; The appendix contains the code and documentation of a Java program for computing in modular lattices. This program led to the discovery and proof of several of the theorems in this work. It may be useful to other researchers in lattice theory.