Algebraic Relativization and Arrow Logic [PhD Thesis]

Contents ......Page 5
Acknowledgments ......Page 7
Introduction ......Page 9
1.1 Arrow logic is the modal logic of transitions ......Page 13
1.2 Cylindric modal logic is the modal logic of assignments ......Page 15
1.4 Fine structure of definability ......Page 18
1.5 BAO’s and general modal logic ......Page 19
2.1 BAO’s, general modal logic and Kripke frames ......Page 21
2.2 Review of basic duality theory ......Page 26
2.3 Relativization and the logical core ......Page 31
2.4 Relation algebras, arrow logic and arrow frames ......Page 33
2.5 Cylindric algebras, cylindric modal logic and alpha frames ......Page 44
3.1 Filtrations ......Page 53
3.2 Relativized relation algebras ......Page 58
3.3 Relativized cylindric algebras ......Page 60
3.4 Concluding remarks ......Page 63
4.1 Axiomatizing BAO’s by representing frames ......Page 65
4.2 Relativized relation algebras ......Page 66
4.3 Reducts of relativized relation algebras ......Page 77
4.4 Adding the difference operator ......Page 82
4.5 Representing BAO’s as algebras of relations ......Page 93
4.6 Concluding remarks ......Page 103
5. Amalgamation & Interpolation ......Page 105
5.1 Amalgamation, interpolation and definability ......Page 106
5.2 Zigzag products ......Page 112
5.3 Preservation ......Page 118
5.4 Applications to relation and cylindric algebras ......Page 121
5.6 Appendix: Reformulation of (S)AP with applications ......Page 126
6.1 Introduction ......Page 135
6.2 The core language ......Page 138
6.3 Expansions of the core language ......Page 141
6.4 Two-Sorted arrow logic ......Page 152
6.5 Concluding remarks ......Page 158
Bibliography ......Page 159
Index ......Page 167
List of symbols ......Page 169
Samenvatting ......Page 173