Adding fuzziness to terms and powerobjects using a monadic approach

The double contravariant powerset monad in the Goguen category of fuzzy sets

“…The triple (F, µ, η) is a submonad of the monad (P, µ, η) in Set(Q), and it is a lifting of the Q-filter monad (Q-Fil, µ, η) in Set along the forgetful functor. 4 The covariant powerset monad in Set(Q)…”

“…Proof. The conclusion is contained in [4], the verification is included here for convenience of the reader.…”

“…The Kleisli category of the monad P has been described in [4]. In the following we describe its Eilenberg-Moore algebras with help of the fact that P is a lifting of P Q .…”

“…The main results include: (i) The category of the Eilenberg-Moore algebras of P is equivalent to the opposite category of Set(Q), hence the Goguen category of fuzzy sets is dually monadic over itself, adding another one to the list of pleasant properties of the Goguen category. (ii) For a commutative quantale, the covariant powerset monad in Set(Q) constructed in [4] is a submonad of the monad P.…”

A survey on the categorical term construction with applications

“…To incorporate fuzziness to terms was considered in [11,10] as a composition of term monad with the many-valued powerset monad L (see [23]) leading to performance of substitution in the Kleisli category Set L•T . The first attempt to construct a term monad over a category of fuzzy sets, namely Set(L), L is a complete lattice, was given in [12], and a construction of a term monad over a sorted category of sets and sorted Set(L) was initiated in [8]. The category Set(L) was essentially determined in [15], thus, referred to as Goguen's category in this paper.…”