Kleisli Monoids Describing Approach Spaces

The saturated prefilter monad

“…As an example, let & be the product t-norm. Since the quantale ([0, 1], &, 1) is isomorphic to Lawvere's quantale ([0, ∞] op , +, 0), it follows that Kleisli monoids and Eilenberg-Moore algebras of the monad (BSP, n, d) are essentially approach spaces [5] and injective approach spaces [12], respectively. As another example, let & be a continuous tnorm that satisfies the condition (S).…”

“…Let & be the product t-norm. Then, the natural transformation n in Equation (7.1) is essentially the multiplication of the monad of functional ideals in [5,Subsection 2.3]. To see this, first we show that for each bounded saturated prefilter F on X and each α > 0,…”

“…Functional ideals play an important role in the theory of approach spaces, see, e.g. [4,5,27,28,29,30]. This example shows that functional ideals are essentially bounded saturated prefilters for the product t-norm.…”

“…Therefore, µ is essentially the map l µ in [5, Subsection 2.2], and consequently, the natural transformation n is the multiplication of the monad of functional ideals in [5,Subsection 2.3].…”

“…The functor of Q-filters in the sense of [7,9] and the functor of Q-filters in the sense of [16,17] are both a monad on the category of sets, see [9,17]. It is shown in [5] that the functor of functional ideals is also a monad on the category of sets. But, it is still unknown whether the functor of prefilters is a monad.…”