Axioms for Lattices and Boolean Algebras

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.

Sample Chapter(s)
Introduction (87 KB)
Chapter 1: Semilattices and Lattices (299 KB)

• Semilattices and Lattices
• Modular Lattices
• Distributive Lattices
• Boolean Algebras
• Further Topics and Open Problems
• Appendices:
  • Some Prover9 Proofs
  • Partially Ordered Sets and Betweenness
  • Quasilattices
  • Lukasiewicz-Moisil Algebras
  • Testing Associativity
  • Complete Existential Theory and Related Concepts