The Chebyshev Set Problem in Riesz Space

In this paper, we mainly study the best approximation theory in Riesz space, which is not constructed by the norm, but only rely on the order structure. Based on the order structure, we propose the concept of the order best approximation in Riesz space and discuss some problems related to the order best approximation, including some sufficient and necessary conditions for satisfying the order best approximation set. Finally, we consider the order best approximation projection and its related properties.