Varieties of Lattices
The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.
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algebras Amal(V amalgamation property Applications Arguesian assume atoms bounded Chapter choose Clearly complemented complete Con(A condition congruence distributive variety Consequently consider construct contains contradiction Conversely Corollary covers define definition denote Desargues distinct distributive lattice dual dually element elementary embedding epimorphism equational equivalent excludes exists extension fact fails Figure filter finite lattice finitely based follows free lattice Geometry give given hence holds homomorphic image identity implies includes isomorphic join irreducible join-cover Jónsson lattice varieties least Lemma length meet minimal modular lattice n-frame nontrivial Note Observe obtain points prime quotient Proceedings projective projective space PROOF prove quotient relations relative representation result Rose satisfies semidistributive sequence similarly splitting lattice steps subdirectly irreducible sublattice subset subvarieties Suppose Theorem Theory trivial unique whence