The ⊤-filter monad and its applications☆
Introduction
Monad (also called the triple) is an endofunctor, together with two natural transformations satisfying the coherence conditions. It has close relation to adjoint functors, every adjunction gives rise to a monad and every monad arises from an adjunction, but in more than one way. Essentially, there are a maximal and a minimal solution to the problem of finding an adjunction from a given monad. In fact, one is the Eilenberg–Moore category and the other one is the Kleisli category. Now, the monadic approaches are very important ways to study topological structures. For example, in [25], Manes showed that the category of Eilenberg–Moore algebras for the ultrafilter monad on the category Set of sets is the category of compact Hausdorff topological spaces and continuous maps. As Barr observed in [1], if one replaces the map between ultrafilters and points by arbitrary relation, then the lax algebras (or relational algebras) with respect to ultrafilter monad can also characterize topological spaces. When the ultrafilter monad is replaced by filter monad, Hofmann and Tholen gave a comprehensive study of topological spaces by Kleisli composition in [15].
Convergence spaces and uniform spaces are two important concepts in mathematics and have close relation to topology. The monadic approaches to these spaces are also very interesting. In fact, following Barr's idea, Clementino, Hofmann and Tholen and Seal showed that many topological structures, such as approach spaces, metric spaces, (quasi-)uniform spaces etc., can all be viewed as lax algebras with respect to certain monads (see [4], [5], [6], [7], [16]). Gähler in [11] studied monadic convergence structures. In [28], Salbany introduced the concept of completion monad and showed that each algebra with respect to completion monad is just the complete quasi-uniform space. In the monograph [16], there is a detailed study of the relationships among topologies, convergence spaces and lax algebras of (ultra)filter monad.
As we know, the concept of filters plays a key role in classical topology. In lattice-valued setting, prefilter, ⊤-filter and L-filter are the three most important kinds of lattice-valued filters. The related topics with these lattice-valued filters draw much attention in fuzzy topology, such as lattice-valued convergence spaces and (quasi)-uniform spaces, etc. (see [8], [13], [14], [17], [21], [23], [24], [27], [30], [31], [32], [34]).
In this paper, we focus on the ⊤-filters. We will generalize the theory of classical filter monads to ⊤-filter setting, and use ⊤-filter monad to study ⊤-convergence spaces and the completion of probabilistic uniform spaces. In Section 3, we introduce ⊤–filter functor and two natural transformations. Under the assumption that being Scott continuous, we show that the triple is a monad and use it to give a characterization of strong L-topologies. In Section 4, we give a lax extension of ⊤-filter monad to Rel—the category of relation—and give representations of topological ⊤-convergence spaces by reflexive and transitive lax algebras. It is shown that the reflexive and unitary lax algebra with respect to ⊤-filter monad exactly is the pretopological ⊤-convergence space, and the reflexive and transitive lax algebra exactly just the topological and pretopological ⊤-convergence space. Hence the category of reflexive and transitive lax algebras of ⊤-filter monad is isomorphic to the category of strong L-topological spaces. Finally, in Section 5, when L is a continuous lattice, we introduce the construction of separated completion monad based on ⊤-filter monad in probabilistic uniform spaces, and prove that a probabilistic uniform space is ⊤-Hausdorff separated complete space if and only if it is an algebra with respect to the separated completion monad. We also study the completion monad and show that the algebras with respect to this completion monad are complete probabilistic uniform spaces.
Section snippets
Preliminaries
A commutative quantale is a pair satisfying that L is a complete lattice with the top element ⊤ and the bottom element ⊥, and ⁎ is a commutative semigroup operation on L such that for all and . For a given commutative quantale , there exists a binary operation defined by called the implication (operation). Further, ⁎ and → form an adjoint pair in the sense of for all .
A commutative quantale is said to
⊤–filter monad
In monoidal topology, the filter monad is a basic tool to study topology and convergence. In this section, we will give the construction of ⊤–filter monad and hope it may play the role of filter monad in lattice-valued topology. First, we list some preliminaries of the ⊤–filter functor and two natural transformations in the following.
(1) Define the ⊤–filter functor as follows:
For each object , ( is usually simplified by TX);
For each , is defined by
Lax extension of ⊤–filter monad
The research of ⊤–convergence spaces draws much attention in fuzzy community. For example, Fang and Yue [8] introduced ⊤–convergence spaces and studied the Kowalsky's diagonal condition and Fischer's diagonal condition. Yu and Fang [31] showed that the category of ⊤–convergence spaces is Cartesian-closed. In [23], weakening the condition of L being a MV-algebra to a meet continuous quantale, Li studied ⊤–convergence spaces associated with CNS spaces. In [27], Reid and Richardson introduced the
Separated completion ⊤–filter monad
In this section, as another application of ⊤–filter monad , we will introduce the construction of separated completion ⊤–filter monad in probabilistic uniform spaces based on ⊤–filter monad, and use separated completion ⊤–filter monad to give an algebraic characterization of ⊤–Hausdorff separated complete probabilistic uniform spaces. In this section, except L is a commutative unital divisible quantale, we also need to assume that L is a continuous lattice. From [12], [29], we know
Acknowledgements
The authors are grateful to the anonymous referees for their valuable comments and suggestions.
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This work is supported by Natural Science Foundation of Shandong Province (ZR2017MA017) and National Natural Science Foundation of China (11471297).