Fuzzy sets and sheaves. Part Ⅱ: Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology

摘要

This essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory, resp. of the theory of complete Ω-valued sets. Hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect. Part Ⅱ of this series of two papers explains the sheaf-theoretic basis of Zadeh's operations on fuzzy sets (with the exception of the complementation). Further, the quotient problem w.r.t. similarity relations and the quotient problem of fuzzy groups w.r.t. fuzzy congruence relations are solved. Finally, stratified fuzzy topologies are identified with internal topologies on constant sheaves.