2.7.1 Partial Orders Video
PROFESSOR: Partial orders are another way to talk about digraphs and they offer to us an interesting lesson in the idea of axiomatizing a mathematical structure and mathematical ideas. So let's begin by discussing some of the properties that we're going to use to axiomatize partial orders and digraphs. So if we think about walks in a digraph, the basic property of walks is that if you have a walk from u to v and you have a walk from v to w, then you put the two walks together and you wind up with a walk from u to w. Expressed in terms of the positive walk relation in G, what this is saying is that if u G plus v and v G plus w, then u G plus w. And that abstract property-- which I'm highlighting with the magenta box-- when you apply it to an arbitrary relation is called a transitivity property. So a relation R on a set-- that is, R is relating-- the domain and co-domain of R are the same-- is that u R v and v R w implies u R w. And a relation that has that prop...