/pdf/a-course-in-universal-algebra-3xkp961xgs.pdf

A Course in Universal Algebra

Journal of the ACM

Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.

https://www.springer.com/cda/content/document/productFlyer/productFlyer_978-3-0348-0017-4.pdf?SGWID=0-0-1297-174034875-0

Lattice Theory: Foundation

http://www.mat.uc.pt/~ct99/abstracts/tholen.pdf

Semi-abelian categories

Introduction. Preliminaries. Basic concepts. Lattices. Unary and binary operations. Fundamental algebraic results. Unique factorization. Bibliography. Table of notation. Index.

/pdf/algebras-lattices-varieties-5sl4cpizfz.pdf

Algebras, lattices, varieties

A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity

The implication and finite implication problems for embedded multivalued database dependencies are both shown to be algorithmically undecidable. The proof is by an interpretation of semigroup word problems via systems of permuting equivalence relations into database dependencies. In contrast, it is shown that for each fixed premise H one has a decision procedure for implications H ? F.

On the Undecidability of Implications between Embedded Multivalued Database Dependencies

S-verklebte Summen von Verbänden

By an implication for database dependencies we mean an expression H ⇒ F , where H is a conjunction of dependencies and F a single dependency. Fixing a class of such implications, a solution of the (finite) implication problem consists in an algorithmic procedure deciding for every implication in the class whether or not it holds in all (finite) databases (in which it is to be interpreted). In [3] this problem was studied for dependencies which are functional (fd) or embedded multivalued (emvd). As pointed out by Luc Segoufin, what was really shown is the following. Theorem 1. The implication problem and the finite implication problem for implicationsH ⇒ F , where F is an emvd and H a conjunction of emvds and fds, are unsolvable.

Corrigendum: Corrigendum to “On the undecidability of implications between embedded multivalued database dependencies” [Inform. and Comput. 122 (1995) 221--235]

Christian Herrmann

Papers

A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity

On the Undecidability of Implications between Embedded Multivalued Database Dependencies

S-verklebte Summen von Verbänden

Corrigendum: Corrigendum to “On the undecidability of implications between embedded multivalued database dependencies” [Inform. and Comput. 122 (1995) 221--235]

On the word problem for the modular lattice with four free generators