Overview

The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of join and meet or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems.A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which according to G Gratzer, a leading expert in modern lattice theory is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices.Contents: Semilattices and LatticesModular LatticesDistributive LatticesBoolean AlgebrasFurther Topics and Open Problems Appendices: Some Prover9 ProofsPartially Ordered Sets and BetweennessQuasilatticesLukasiewicz-Moisil AlgebrasTesting AssociativityComplete Existential Theory and Related ConceptsReadership: Advanced undergraduate and graduate students as well as researchers in mathematics and computer science; algebraists; logicians; software designers (especially designers of theorem provers); theoretical computer scientists; people in the industry of artificial intelligence, automated reasoning and robotics.

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9781299604872

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