In topology, two structures are equivalent if, and only if, conserved under a transformation of one structure sending its open sets into open sets of the other. This "sending open sets into open sets" is the basic topological transformation.

We need an ordinological transformation that defines the field.

There is only ONE CONDITION for an ORDER-STRUCTURE: TRANSITIVITY. I can easily graph TRANSITIVITY by using three different COLORED ARROWS for THREE PATHWAYS:  Given ÞÞ   and given also   ÞÞ   then TRANSITIVITY means   ÞÞÞ.

For COMPUTING, you often need a TRANSITIVE PATH from one INFO-BIT to ANOTHER. The full COMPLETION may be done by many PATHS. ThE MINIMAL PATH is known as TRANSITIVE CLOSURE. In general, it is subtle to verify. But there is an ALGORITHM to do so.

The basic problem of transitive closure is "how to get there from here" and do so "by the shortest path".

I'll cite a fundamental problem in computer science of "yesteryear". Remember the "Y2K" problem discussed so much in the years 1998-9?

Basic software from the 70's was used to build new software. Back then, the new millenium seemed "far away". So, to conserve "space", the software allowed only changes in the last two digits of the four-digit specification of the current year. Thus, you could change, say, 1987 to 1988 or even 1989 to 1990. But you could not change 1999 to 2000.

When the "Y2K" problem began to be well known to the general public, some of my friends with computers could not understand why a problem existed. "I know where the little program is to call up for changing the date. Why can't they find it, too?"

I tried to explain to them that they were conversing with the computer in English ("date") and decimal numeration. But the computer really does not speak English and especially avoids decimal numeration as much as possible, preferring binary numeration. Computer language is basically about "off" and "on" of "switches" and the mediary language between computerese and humanese is coding in terms of "0" and "1".

So, the basic problem is "how to get there from here": to get from "frontend" talk to the disk's "foreign talk". And, to cut overtime, how to "get there by the shortest path". In other words, transitive closure. Dig?

I propose transitive closure as the defining transformation for Ordinology. If you can improve upon my treatment of it -- or propose a better defining transformation -- be my guest.