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

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.