Axioms for Lattices and Boolean Algebras.
Başlık:
Axioms for Lattices and Boolean Algebras.
Yazar:
Padmanabhan, R.
ISBN:
9789812834553
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 online resource (228 pages)
İçerik:
Contents -- Introduction -- 1. Semilattices and Lattices -- 1.1. Semilattices -- 1.2. Defining Lattices in Terms of the Operations _ and ^ -- 1.3. One-Based Theories -- 1.4. Defining Lattices by Other Tools -- 2. Modular Lattices -- 2.1. Modular Lattices within Lattices -- 2.2. Defining Modular Lattices in Terms of the Operations v and ^ or by Other Tools -- 2.3. Self-Dual Equational Bases for Modular Varieties -- 3. Distributive Lattices -- 3.1. Distributive Lattices within Lattices -- 3.2. Defining Distributive Lattices in Terms of the Opera- tions v and ^ -- 3.3. Defining Bounded Distributive Lattices in Terms of the Median Operation -- 4. Boolean Algebras -- 4.1. Boolean Lattices -- 4.2. Boolean Algebras in Terms of the Operations v -- ^ and 0 -- 4.3. Boolean Algebras in Terms of the Operations v and 0 -- 4.4. Boolean Rings and Groups -- 4.5. Boolean Algebras in Terms of Nonassociative Binary Operations -- 4.6. Boolean Algebras in Terms of Ternary or n-ary Operations -- 4.7. Boolean Algebras in Terms of Relations -- 4.8. Huntington Varieties -- 4.9. Boolean Algebras Are Always One-Based -- 4.10. Orthomodular Lattices -- 5. Further Topics and Open Problems -- I. Tarski-type theorems on independent equational bases -- II. Huntington varieties of lattices -- III. Binary reducts of Boolean algebras -- IV. Frink-type theorems for varieties of complemented lattices -- Appendix A: Some Prover9 Proofs -- References -- Appendix B: Partially Ordered Sets and Betweenness -- References -- Appendix C: Quasilattices -- Appendix D: Lukasiewicz-Moisil Algebras -- Appendix E: Testing Associativity -- Appendix F: Complete Existential Theory and Related Concepts -- Bibliography -- Index.
Özet:
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.
Notlar:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Tür:
Yazar Ek Girişi:
Elektronik Erişim:
Click to View