Abstract
Abstract In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of convex spaces is isomorphic to the category HS of hull spaces, and they are all topological over Set. Also, it is proved that there is an adjunction between the category IS of interval spaces and the category CS of convex spaces. In particular, the category CS(2) of arity 2 convex spaces can be embedded in IS as a reflective subcategory.
Highlights
Convexity is an important and basic property in many mathematical areas
We introduced the axiomatic conditions of hull operators and showed that the resulting category is isomorphic to the category of convex spaces
We investigated the relations between convex spaces and interval spaces
Summary
Convexity is an important and basic property in many mathematical areas. in some concrete mathematical setting, such as vector spaces, it is not the most suitable setting for studying the basic properties of convex sets. For spaces derived from an algebraic structure, convexity-preserving mappings usually agree with the corresponding notions of homomorphisms. There are close relations between convex structures and interval operators. A mapping f : (X, CX) −→ (Y , CY ) is called convexity-preserving (CP, in short) provided that B ∈ CY implies f ←(B) ∈ CX.
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