A short note on L-fuzzy approximation spaces and L-fuzzy pretopological spaces
Introduction
The rough set theory was originally proposed by Pawlak [11], [12] as a mathematical approach for handling imprecision and uncertainty in data analysis. In recent years, rough set theory has developed significantly due to its widespread applications. Various generalized rough set models have been established and their properties or structures have been investigated intensively [22], [23].
An interesting and natural research topic in rough set theory is the study of rough set theory via topology. In 1988, Skowron [17] and Wiweger [21] separately addressed this topic using classical rough set theory. In 1994, Kortelainen [7] considered the relationship between modified sets, topological spaces, and rough sets based on a preorder. Subsequently, in 2005, as generalizations of rough sets from the viewpoint of fuzzy sets, Qin and Pei [14] showed that a one-to-one correspondence exists between the family of all the lower approximation sets based on fuzzy preorder and the set of all fuzzy topologies that satisfy the so-called (TC) axiom. In 2011, Pei et al. [13] observed that inverse serial relations are the weakest relations that can induce topological spaces, and that different relations based on generalized rough set models will induce different topological spaces. In addition, Hao and Li [5] determined a one-to-one correspondence between the set of all reflexive, transitive L-fuzzy relations and the set of all Alexandrov L-fuzzy topologies. In 2013, Ma and Hu [10] investigated the topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets.
However, the relationship between L-fuzzy pretopological spaces and L-fuzzy approximation spaces is not clear. For a binary L-fuzzy relation, we have an L-fuzzy approximation space, so we can then obtain a lower approximation operator. Furthermore, we can obtain a category composed of objects of L-fuzzy approximation spaces. Hence, in this note, we investigate the relationship between L-fuzzy pretopological spaces and L-fuzzy approximation spaces based on the reflexive L-fuzzy relations from a categorical viewpoint.
The remainder of this paper is organized as follows. In Section 2, we recall some fundamental concepts and related properties of complete residuated lattices and adjoint functors, which are needed in the sequel. In Section 3, we mainly discuss the relationship between L-fuzzy pretopological spaces and L-fuzzy approximation spaces based on the reflexive L-fuzzy relations. In the final section, we give the conclusions of our study.
Section snippets
Preliminaries
A complete residuated lattice is a pair subject to the following conditions:
- (1)
L is a complete lattice with a top element 1 and a bottom element 0;
- (2)
is a commutative monoid;
- (3)
for all and .
For the convenience of readers, we provide some basic properties of the operations on complete residuated lattices in the following proposition.
Proposition 2.1 (See Blount and Tsinakis
Main results
An L-fuzzy relation on a set X is a map . Let R be an L-fuzzy relation on X. Then, R is reflexive if for all x in X and R is &-transitive if for all . An L-fuzzy relation R is called an L-fuzzy preorder if it is reflexive and &-transitive [2], [4], [8].
Definition 3.1 (See Höhle and Šostak [18], Zhang [25].) An L-fuzzy topology on a set X is a subset such that . for all . for every family . An L-fuzzy topology η is strong if it
Conclusions
In this short note, we mainly obtained an adjoint between the category of L-fuzzy approximation spaces based on the reflexive L-fuzzy relations and the category of strong L-fuzzy pretopological spaces. We also showed that the adjoint obtained in this note is an extension of the adjoint between the category of L-fuzzy approximation spaces based on L-fuzzy preorders and the category of strong L-fuzzy topological spaces.
Acknowledgements
This study was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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