Preface ....................................................... VII
Chapter I Introduction .......................................... 1
1 History ...................................................... 1
2 Survey of Results ............................................ 9
Chapter II Algebra ............................................. 19
1 Universal Algebra ........................................... 19
2 Products and Factor Objects ................................. 35
Chapter III Tools .............................................. 55
1 Model Theory ................................................ 55
2 Category Theory ............................................. 59
3 Topology .................................................... 72
4 Boolean Algebras ............................................ 74
Chapter IV Complexes and their Sheaves ......................... 79
1 Concepts .................................................... 80
2 Constructions ............................................... 87
3 Categorical Reformulation ................................... 96
Chapter V Boolean Subsemilattices ............................. 109
1 Identifying the Congruences ................................ 110
2 Constructing the Complex ................................... 116
3 Special Sheaves ............................................ 123
4 Categorical Recapitulation ................................. 130
Chapter VI Sheaves from Factor Congruences .................... 147
1 Factorial Braces ........................................... 148
2 Boolean Algebras of Factor Objects ......................... 153
3 Algebras Having Boolean Factor Congruences ................. 165
4 Their Categories ........................................... 176
Chapter VII Shells ............................................ 179
1 Algebras with a Multiplication ............................. 181
2 Half-shells ................................................ 186
3 Shells ..................................................... 195
4 Reprise .................................................... 207
5 Separator Algebras ......................................... 212
6 Categories of Shells ....................................... 216
Chapter VIII Baer-Stone Shells ................................ 221
1 Integrality ................................................ 222
2 Regularity ................................................. 231
Chapter IX Strict Shells ...................................... 235
1 Nilpotents and Null-symmetry ............................... 236
2 Converses and Axiomatics ................................... 246
3 Adding a Unity or a Loop ................................... 252
Chapter X Varieties Generated by Preprimal Algebras ........... 261
1 Overview ................................................... 261
2 From Permutations .......................................... 264
3 From Groups ................................................ 266
4 From Subsets ............................................... 268
5 Remaining Preprimal Varieties .............................. 269
Chapter XI Return to General Algebras ......................... 273
1 Iteration .................................................. 273
2 Self Help .................................................. 278
Chapter XII Further Examples Pointing to Future Research ...... 283
1 From Classical Algebra ..................................... 283
2 Algebras from Logic ........................................ 285
3 From Model Theory .......................................... 288
4 Beyond Sheaves over Boolean Spaces ......................... 290
5 Many Choices ............................................... 291
List of Symbols ............................................... 295
References .................................................... 301
Index ......................................................... 319
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