Construction of K_α-orders including admissible ones on classes of discrete intervals

Ordering relations such as total orders, partial orders and preorders play important roles in a host of applications such as automated decision making, image processing, and pattern recognition. A total order that extends a given partial order is called an admissible order and a preorder that arises by mapping elements of a non-empty set into a poset is called an h-order or reduced order. In practice, one often considers a discrete setting, i.e., admissible orders and h-orders on the class of non-empty, closed subintervals of a finite set L_n={0,1,….,n}. We denote the latter using the symbol I_n^⁎. Admissible orders and h-orders on I_n^⁎ can be generated by the function that maps each interval x=[x_,x‾]∈I_n^⁎ to the convex combination K_α(x)=(1−α)x_+αx‾ of its left and right endpoints. In this paper, we determine a set consisting of a finite number of relevant α's in [0,1] that generate different h-orders on I_n^⁎. For every n∈N, this set allows us to construct the families of all h-orders and admissible orders on I_n^⁎ that are determined by some convex combination. We also provide formulas for the cardinalities of these families in terms of Euler's totient function.