1 Exotic equations ............................................. 1
1 An abstract theory of equations ........................... 1
2 Equations over finite sets ................................ 5
2 Universal algebra ........................................... 13
1 First concepts and subdirect decompositions .............. 13
2 Term algebra, identities and polynomials ................. 17
3 Polynomials, identities (continued) and algebraic
functions ................................................ 22
3 Lattices .................................................... 31
1 Posets and distributive lattices ......................... 31
2 Classes of (relatively) (pseudo)complemented lattices .... 35
3 Functions and equations .................................. 47
4 Equational compactness of lattices and Boolean algebras ..... 61
1 Abstract equational compactness .......................... 61
2 Equational monocompactness of semilattices and
lattices ................................................. 62
3 Equational compactness of Boolean algebras ............... 66
5 Post algebras ............................................... 69
1 Basic properties of Post algebras ........................ 69
2 Post functions ........................................... 83
3 Post equations ........................................... 90
6 A revision of Boolean fundamentals ......................... 125
1 Linear Boolean equations ................................ 125
2 Generalized minterms and interpolating systems .......... 134
3 Prime implicants and syllogistic forms .................. 141
4 Reproductive solutions, recurrent inequalities and
recurrent covers ........................................ l53
5 Generalized systems/solutions of Boolean equations ...... 167
7 Closure operators on Boolean functions ..................... 177
1 A general theory ........................................ 177
2 Isotone and monotone closures ........................... 181
3 Independent and decomposition closures .................. 192
8 Boolean transformations .................................... 209
1 Functional dependence of Boolean functions .............. 209
2 The range of a Boolean transformation ................... 216
3 Injectivity domains of Boolean transformations .......... 219
4 Fixed points of lattice and Boolean transformations ..... 225
9 More on solving Boolean equations .......................... 231
1 Special methods for solving Boolean equations ........... 231
2 Boolean equations with unique solution .................. 249
3 Quadratic truth equations ............................... 253
4 Boolean equations on computers .......................... 264
10 Boolean differential calculus .............................. 267
1 An informal discussion .................................. 267
2 An axiomatic approach ................................... 274
3 Boolean differential equations .......................... 282
11 Decomposition of Boolean functions ......................... 289
1 A historical sketch ..................................... 289
2 Decomposition via Boolean equations ..................... 294
12 Boolean-based mathematics .................................. 303
1 Mathematical logic ...................................... 303
2 Post-based algebra ...................................... 312
3 Geometry ................................................ 314
4 Statistics .............................................. 324
13 Miscellanea ................................................ 329
1 Equations in MVL and relation algebras .................. 329
2 Equations in functionally complete algebras ............. 333
3 Generalized Boolean functions and non-Boolean
functions ............................................... 337
4 Functional characterizations of classes of functions
over В .................................................. 344
5 Local properties of Boolean functions and extremal
solutions of Boolean equations .......................... 349
14 Applications ............................................... 359
1 Graph theory ............................................ 359
2 Automata theory ......................................... 365
3 Synthesis of circuits ................................... 372
4 Fault detection in combinational circuits ............... 380
5 Databases ............................................... 383
6 Marketing ............................................... 386
7 Other applications ...................................... 389
Appendix 1 Errata to BFE ..................................... 395
Appendix 2 Decomposition of Boolean functions and
applications: a bibliography ...................... 397
Appendix 3 Open problems ..................................... 405
Bibliography .................................................. 429
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