To explain and develop the title (which contains two words of my own creation), I must first review the model that inspired me.

In the field of combinatorial topology, one builds up a structure from the gnomon of a simplex.

Definition 1: An n-simplex is an n-dimensional structure composed of n+1 simplices of order n = 0, 1, 2, ....

Definition 2: The convexity of spatial elements of order s is the extension of all space between these elements.

Thus,

• a 0-simplex is a point;
• a 1-simplex is a line-segment, the convexity of two 0-simplices;
• a 2-simplex is an equilateral triangle, the convexity of three 1-simpices;
• a 3-simplex is a regular tetrahedron, the convexity of four 2-simplices;
• etc.

One then builds a complex from one or more simplices. Note that these are polygonal structures or have polygonal components. (The purpose is to make use of Euler's Theorem which applies to the vertices, edges, faces of polygons.)

---

Modeling the word "polygon", I create the word "ordonal" for a structure of ordinology. I shall build an ordonal from the gnomon of ordex.

In ordinology, we consider all forms of ordering, many of which are listed in the file, "ordinology:, at this Website. Most of our files have concerned only total or simple ordering and partial ordering. And we have described three structures capable of being designated as ordexes. This number can be expanded:

• a transition, representing the transitivity of any ordering, hence, the most elemental gnomon, designated as an e-ordex (since "t" is reserved for "type");
• a hierarch -- parordering in most elemenary form -- which we can designate as an h-ordex;
• the bierarch -- hierarch with binomial "face" -- which we can designate as a p-ordex;
• the simchain -- simple (a.k.a. total) ordering -- which we can designate as a an i-ordex;
• the gopain -- a simchain with geometric progression "face" -- which we can designate as an g-ordex.
• a sort, a transition operating on a transition, which we can designate as an s-ordex;