{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/9054"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/9054","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"Congruences and amalgamation in small lattice varieties","abstract":"When it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are finitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of finitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under finite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a finitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a finite dimensional simple lattice.","abstract_html":"When it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are finitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of finitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under finite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a finitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a finite dimensional simple lattice.","abstract_has_math":false,"creators":["Ouwehand, Peter"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Rose, Henry"],"committee_chairs":[],"committee_members":[],"year":1998,"date_issued":"1998","date_published":"1998","updated_at":"2026-07-22T22:23:21Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/9054","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Rose, Henry"]},{"key":"dc:creator","label":"Author","values":["Ouwehand, Peter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-11-04T08:43:36Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-11-04T08:43:36Z"]},{"key":"dc:date.issued","label":"Date","values":["1998"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Doctoral Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/9054"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Bibliography: pages 108-110."]},{"key":"dc:description.abstract","label":"Abstract","values":["When it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are finitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of finitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under finite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a finitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a finite dimensional simple lattice."]},{"key":"dc:title","label":"Title","values":["Congruences and amalgamation in small lattice varieties"]}]}],"canonical_facts":{"dc:contributor.advisor":["Rose, Henry"],"dc:creator":["Ouwehand, Peter"],"dc:date.accessioned":["2014-11-04T08:43:36Z"],"dc:date.available":["2014-11-04T08:43:36Z"],"dc:date.issued":["1998"],"dc:description":["Bibliography: pages 108-110."],"dc:description.abstract":["When it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are finitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of finitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under finite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a finitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a finite dimensional simple lattice."],"dc:identifier.uri":["http://hdl.handle.net/11427/9054"],"dc:language.iso":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:title":["Congruences and amalgamation in small lattice varieties"],"dc:type":["Doctoral Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-22T22:23:21Z"}