Lattice Functions and Equations

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.

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.