Lattice Functions and Equations

Lattice Functions and Equations
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Artikel-Nr:
9781852332662
Veröffentl:
2001
Einband:
Paperback
Erscheinungsdatum:
30.07.2001
Seiten:
452
Autor:
Sergiu Rudeanu
Gewicht:
680 g
Format:
235x155x24 mm
Serie:
Discrete Mathematics and Theoretical Computer Science
Sprache:
Englisch
Beschreibung:

Lattice (Boolean) functions are algebraic functions defined over an arbitrary lattice (Boolean algebra), while lattice (Boolean) equations are equations expressed in terms of lattice (Boolean) functions.This self-contained monograph surveys recent developments of Boolean functions and equations, as well as lattice functions and equations in more general classes of lattices; a special attention is paid to consistency conditions and reproductive general solutions.The contents include:- equational compactnessin semilattices and Boolean algebras;- the theory of Post functions and equations (which is very close to that of Boolean functions and equations);- a revision of Boolean fundamentals;- closure operators on Boolean functions;- the decomposition of Boolean functions;- quadratic truth equations;- Boolean differential calculus;- Boolean geometry and other topics.There is also a chapter on equations in a very general sense. Applications refer to graph theory, automata theory, synthesis of circuits, fault detection, databases, marketing and others.
1. Exotic equations.- 1 An abstract theory of equations.- 2 Equations over finite sets.- 2. Universal algebra.- 1 First concepts and subdirect decompositions.- 2 Term algebra , ident ities and polynomials.- 3 Polynomials, identities (continued) and algebraic functions.- 3. Lattices.- 1 Posets and distributive lattices.- 2 Classes of (relatively) (pseudo)complemented lattices.- 3 Functions and equations.- 4. Equational compactness of lattices and Boolean algebras.- 1 Abstract equational compactness.- 2 Equational monocompactness of semilattices and lattices.- 3 Equational compactness of Boolean algebras.- 5. Post algebras.- 1 Basic properties of Post algebras.- 2 Post functions.- 3 Post equations.- 6. A revision of Boolean fundamentals.- 1 Linear Boolean equations.- 2 Generalized minterms and interpolating systems.- 3 Prime implicants and syllogistic forms.- 4 Reproductive solutions , recurrent inequalities and recurrent covers.- 5 Generalized systems/solutions of Boolean equations.- 7. Closure operators on Boolean functions.- 1 A general theory.- 2 Isotone and monotone closures.- 3 Independent and decomposition closures.- 8. Boolean transformations.- 1 Functional dependence of Boolean functions.- 2 The range of a Boolean transformation.- 3 Injectivity domains of Boolean transformations.- 4 Fixed points of lattice and Boolean transformations.- 9. More on solving Boolean equations.- 1 Special methods for solving Boolean equations.- 2 Boolean equations with unique solution.- 3 Quadratic truth equations.- 4 Boolean equations on computers.- 10. Boolean differential calculus.- 1 An informal discussion.- 2 An axiomatic approach.- 3 Boolean differential equations.- 11. Decomposition of Boolean functions.- 1 A historical sketch.- 2 Decomposition via Boolean equations.- 12. Boolean-based mathematics.- 1 Mathematical logic.- 2 Post-based algebra.- 3 Geometry.- 4 Statistics.- 13. Miscellanea.- 1 Equations in MVL and relation algebras.- 2 Equations in functionally complete algebras.- 3 Generalized Boolean functions and non-Boolean functions.- 4 Functional characterizations of classes of functions over B.- 5 Local properties of Boolean functions and extremal solutions of Boolean equations.- 14. Applications.- 1 Graph theory.- 2 Automata theory.- 3 Synthesis of circuits.- 4 Fault detection in combinational circuits.- 5 Databases.- 6 Marketing.- 7 Other applications.- Appendix 1. Errata to BFE.- Appendix 2. Decomposition of Boolean functions and applications: a bibliography.- Appendix 3. Open problems.

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