5.13 Strict orders and directed acyclic graphs
strict order, A relation R if R is transitive and anti-reflexive
directed acyclic graph, is a directed graph that has no positive length cycles.
Topological, is an ordering of the vertices that is consistent with the edges in the graph
Overview:
On the elements of its domain, a partial order behaves similarly to the operator. On the items of its domain, a strict order behaves similarly to the operator. If R is transitive and anti-reflexive, it is a strict order. aRb is written with the notation a b, which means tta is less than b.tt The tiny curvature of the symbol distinguishes it from the other symbols. A strictly ordered set is denoted by (A, ) and includes the domain as well as the rigorous order specified on it. A rigorous order arrow diagram is essentially a partial order arrow diagram minus the self-loops. The definitions for partial orders naturally transfer over to strict orders. If x y or y x, two elements, x and y, are said to be similar. The elements are said to be incomparable if they arenfit. If every pair of items is comparable, a stringent order is a total order. If there is no y such that y x, an element x is a minimum element.