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.
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Book Description
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.
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defining lattices, lattice betweenness, nonassociative binary operations, uniquely complemented lattice, majority polynomial, equational bases, orthomodular lattices, lattice reduct, equational basis, lattice identities, splitting lattice, bounded distributive lattices, absorption identities, implication algebras, median operation, definitional equivalence, complemented lattices, ternary operation, single axiom, bounded lattice, equational theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Terms of the Operations, Other Tools, Terms of Ternary, One-Based Theories, Sholander's Theorem, Terms of the Median Operation, Terms of Nonassociative Binary Operations, Terms of Operations, Huntington Varieties, Absorption Laws Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
defining lattices, lattice betweenness, nonassociative binary operations, uniquely complemented lattice, majority polynomial, equational bases, orthomodular lattices, lattice reduct, equational basis, lattice identities, splitting lattice, bounded distributive lattices, absorption identities, implication algebras, median operation, definitional equivalence, complemented lattices, ternary operation, single axiom, bounded lattice, equational theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Terms of the Operations, Other Tools, Terms of Ternary, One-Based Theories, Sholander's Theorem, Terms of the Median Operation, Terms of Nonassociative Binary Operations, Terms of Operations, Huntington Varieties, Absorption Laws Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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