In the file, "ORDEX", I've posted a list of ordexes, that is, gnomons of some of the most useful types of orderings:

• 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;
• a 2-chain, a special form of simchain which is a component of an general lattices;
• a 3-chain, a special form of simchain used to compose modular and nonmodular lattices;
• the gopain, a special for of simchain with geometric progression ranking, which we can designate as an g-ordex.

We can now form a list of transformations on orderings:

• a sort is a transition operating on a transition, which we can designate as an s-ordex;
• the p-sort transforms a bierarch into a simchain;
• the aug operator transforms 2-chains into an o-lattice (distributive lattice);
• the c-product operator transforms n-chains into modular and nomodular lattices.
• the reduction operator which transforms a distributive lattice into a modular lattce by "coalescing" two or more elements;
• its inverse, the expansion operator which transforms a nonmodular into a modular lattice, or a modular into a distributive lattice by inserting one or more elements.