On a categorical generalization of the concept of fuzzy set [Elektronische Ressource] / Sergejs Solovjovs

ONACATEGORICAL GENERALIZATION OF THECONCEPT OF FUZZY SETSergejs SolovjovsDissertationzur Erlangung des Grades eines Doktorsder Naturwissenschaften– Dr. rer. nat. –Vorgelegt im Fachbereich 3 (Mathematik & Informatik)der Universitat¨ Bremenim Juni 2007Datum des Promotionskolloquiums: 22. Juni 2007Erster Gutachter: Prof. Dr. Hans-Eberhard Porst (Universitat¨ Bremen)ˇZweiter Gutachter: Prof. Dr. Alexander Sostak (Universitat¨ Lettlands)ContentsAbstract vAcknowledgements viiIntroduction 11 Categories of lattice-valued sets as categories of arrows 51.1 The category X(A)ofA-valuedobjects.................... 51.1.1 Someexamples.................. 71.1.2 Basicsubcategories............................ 91.2 Topological properties of X(A)............. 10∗ ◦1.3 A relation between the functors (−) and (−) ................ 182 On a generalization of Goguen’s category Set(L) 212.1 The category X(A)anditstopologicalproperties............... 212.2 An example of a non-topological category X(A)........ 232.3 On concrete cartesian closedness of X(A)................... 262.4 X(A)isnotatopos ....................... 282.5 On representability of partial morphisms in X(A)............... 292.6 X(A)isaquasitopos....................... 322.6.1 The inner structure of X(A)................... 322.6.2 ArelationbetwenthestructuresgeneratedbyΩandΔ... 362.7 Some remarks on representability of partial morphisms in X(A)....... 373 Aspects of comma categories 41∗3.