The logic of categories of partial functions and its applications

Adam Obtułowicz

GDML_Books, (1986), p.

Résumé

CONTENTS0. Introduction.......................................................................................................................................................................................51. Preliminaries.....................................................................................................................................................................................92. Relations and functional relations in a category.............................................................................................................................13 2.1. The concept of a regular category.............................................................................................................................................13 2.2. Relations in a category and their composition............................................................................................................................14 2.3. Functional relations in a category...............................................................................................................................................15 2.4. Certain generalization of the concept of a category of functional relations................................................................................203. The category $Pfn_A$ as a category with ordered hom-sets.........................................................................................................23 3.1. The concept of an ordered category..........................................................................................................................................23 3.2. The concept of a domain classifier w.r.t. partial ordering............................................................................................................24 3.3. The concept of a product w.r.t. partial ordering and domain classifier........................................................................................264. The category $Pfn_A$ as a category with additional equational structure.....................................................................................29 4.1. The concept of a category with ordered precartesian structure.................................................................................................29 4.2. The concepts of an equoidal category and a semilogical category............................................................................................35 4.3. The notion of the category associated to a category with ordered strict precartesian structure.................................................41 4.4. Functors between categories with ordered strict precartesian structure and the notion of quasi-natural transformation...........475. Functional relations in an elementary topos...................................................................................................................................54 5.1. The notion of an elementary topos.............................................................................................................................................54 5.2. Higher-order types of functionality in $Pfn_E$...........................................................................................................................59 5.3. The notion of a doctrine of functional relations..........................................................................................................................64 5.4. The interpretation of logical connectives in an equoidal category with types and in a doctrine of functional relations...............72 5.5. Undefined elements and upper bounds......................................................................................................................................806. Axiom of infinity. programmability. and recursiveness.....................................................................................................................88 6.1. Axiom of infinity and the notion of an arithmetical doctrine of functional relations......................................................................88 6.2. Programmability in doctrines of functional relations...................................................................................................................96 6.3. Recursiveness and doctrines of functional relations................................................................................................................1027. Applications to Universal Algebra; theories classifying partial algebras........................................................................................105 7.1. The formulation of P-equational logic.......................................................................................................................................105 7.2. The presentation of p-theories by categories...........................................................................................................................111 7.3. The representations of partial algebras by functors.................................................................................................................115 7.4. Categories of partial algebras; p-algebraic categories and p-algebraic functors......................................................................119 7.5. Properties of p-algebraic categories and p-algebraic functors.................................................................................................124 7.6. Characterization of p-algebraic categories...............................................................................................................................131 7.7. Applications in linguistics..........................................................................................................................................................138 7.8. Graphical algebras...................................................................................................................................................................141Appendix A.......................................................................................................................................................................................148Appendix B.......................................................................................................................................................................................149Appendix C.......................................................................................................................................................................................155References.......................................................................................................................................................................................157Index.................................................................................................................................................................................................160Index of symbols...............................................................................................................................................................................163

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Zbl 0617.03041

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