Transformations on General Partial Orders

Abstract

Next we will be considering repetitions of items in the input sequential data, and as a consequence, the final closed partial orders are not necessarily injective. As we will see, dealing with general partial orders makes the proper formalization with category theory a bit more difficult. To start with, we are forced to drop the injectivity of the morphisms in the general category of graphs, and this allows for many different ways of mapping a partial order over a sequence. Still another inconvenience, Theorem 5.1 of chapter 5 just holds for one of the directions in this new problem. Indeed, the maximal paths of the final closed partial orders do not necessarily coincide exactly with the intersections of the compatible input sequential data. Here we will try to formally justify the construction of our final closed partial orders for a set of data as the colimit transformation on path-preserving edges. Colimits will naturally generalize also the coproduct transformation of chapter 5, but as we shall see, proving that our final structure has the property of being maximally specific is still an unsolved combinatorial problem.