Lattice-valued interval operators and its induced lattice-valued convex structures

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Abstract
Galois correspondence in category theory plays an important role in establishing the relationships between different types of spatial structures. In this paper, we apply Galois correspondence as a tool to the theory of lattice-valued convex structures. We mainly introduce the concept of lattice-valued interval operators and discuss its relationships with L-fuzzifying convex structures and L-convex structures. It is shown that there is a Galois correspondence between the category of lattice-valued interval spaces and the category of L-fuzzifying convex spaces. In particular, the category of arity 2 L-fuzzifying convex spaces can be embedded in the category of lattice-valued interval spaces as a reflective subcategory. Also, it is proved that there is a Galois correspondence between the category of lattice-valued interval spaces and the category of L-convex spaces. Specially, the category of arity 2 L-convex spaces can be embedded in the category of lattice-valued interval spaces as a reflective subcategory.
Funding Information
  • The National Natural Science Foundation of China (11371002)
  • the Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048)
  • the Scientific Research Foundation of CUIT (KYTZ201631, CRF201611, 17ZB0093)

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