Fuzzy Sets from a Meta-system- theoretic Point of View

In this paper we present, for the first time, complete proofs of the facts that have already been announced in [A. Daneshgar, A. Hashemi, A general model for I/O system theory, Proc. AIMC31, 2000, 292–299]. Our approach here is to focus on the aspects related to fuzzy set theory and we leave other connections to other mathematical disciplines such as quantum groups, categorical logic, homological algebra and measure theory to appear elsewhere. The main contribution is to introduce a general framework based on enriched category theory that covers the theory of translation invariant systems as a special case, as well as the construction of Haar fraction on a locally compact group.