Amalgamation in varieties of algebras - Jacobs, David Frank (Department of Mathematics and Applied Mathematics, 2016)
One of the most successful approaches to research in universal algebra has been the study of varieties, initiated by Garett Birkhoff in the 1930's. Examples of varieties include many classes of algebras such as groups, semigroups, lattices and Boolean algebras. In 1927, O. Schreier showed that for any set of extensions of a given group, there is another extension of that group that in some sense contains all other extensions in the set. This property of groups, known as the amalgamation property was generalized to a universal-algebraic setting by R. Fralsse in 1954, and the important question as to which varieties satisfied the amalgamation property arose. While some of the answers to this question were positive (such as for the varieties of lattices, distributive lattices and Boolean algebras), many common varieties such as the variety of semigroups and all non-distributive modular lattice varieties were shown to fail to satisfy the amalgamation property. In the light of these negative results, attempts were made to "localize" this property from the variety to its individual members, the most successful being the notions of amalgamation base and amalgamation class, first introduced by George Gratzer and Henry Lakser in 1971. Investigations into the nature of the amalgamation classes of varieties that fail to satisfy the amalgamation property were carried out in the 1970's and 1980's by among others, Clifford Bergman and Henry Rose, the main focus being congruence distributive varieties, of which lattice varieties form the prime example. The topic of amalgamation has also been studied in fields as diverse as topology, logic and the theory of field extensions. In this dissertation, however, I will focus on the more algebraic results concerning amalgamation. My aim is to present a selection of these results, using as examples varieties of groups, semigroups, lattices and Heyting algebras, in a universal-algebraic framework that is (more or less) self-contained and uniform in its notation. Bibliography: pages 142-150.