Approximating SP-orders through total preorders: incomparability and transitivity through permutations

Abstract

We study finite partial orders and the concept of indistinguishability. In particular, we focus on SP-orders. These orderings can be represented by means of Hasse diagrams and numerical labels. Since these numerical representations can be interpreted by means of permutations, we extend the study to the field of group theory. Through this point of view, we introduce the new concept of total extension and total inclusion of a partial order as the total preorders closest to the initial partial order from below and from above, respectively. Finally, we show a possible study of finite T₀ topologies by means of its corresponding partial order.