Fuzzy presubsets as non-idempotent and non-commutative classifications of subalgebras

Authors:
Ulrich Höhle
Affiliations:
Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Gaußstraße 20, D-42097 Wuppertal, Germany
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

In the framework of quantale preordered sets regular Q-fuzzy presubsets are identified with classifiable subalgebras. This identification is based on the monad T of quasi-presingletons which is an extension of the monad associated with the Cauchy completion of quantale preordered sets. Even though not every T-subalgebra is classifiable, the classifiable hull of T-subalgebras exists always. An application of these constructions is the internalization of the negation of the underlying quantale as truth arrow in the category of T-algebras.