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E-raamat: Lattice Functions and Equations

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Lattice Functions and EquationsSuurem pilt
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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 compactness in 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.

Muu info

Springer Book Archives
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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