I created the label "gopain" as ellipsis on three words "geometric progression chain". Thus, g(e)o(metric)p(rogressionc)ain.The guiding model is the factor lattice. As in the factors of 30 = 2*3*5. This has the factor set: {1,2,3,5,6,10.15.30}. This factor set can be assigned elements of a Hasse diagram as follows.
Please note that this t-lattice has the hierarchical structure used in bureaucracies and military services; and the FILE numbers, namely, {1, 3, 3, 1} match those in the third row of a binomial table. For these reasons, I've created a new label, elliptically from two labels. I call the above structure a bierach: b(inomialh)ierarch(y). Yje Hasse diagram of every t-lattice fits the bierarch pattern. But we shall now find that the Hasse diagram of a proper o-lattice is part bierarch pattern, and partly some other pattern.
- You start with the prime factor set of 30, namely, {2, 3, 5}. These are mutually independent, containing only the universal factor of 1. So they can be graphed in partial ordering -- none dominating any other. These form the "atoms" (join-irreducible elements of Rank 1) in the lattice:
2 3 5 (ATOMS, RANK = 1)- Given these, we can introduce the Hasse graph for the greatest common denominator of the atoms , namely, GCD(2, 3, 5) = 1, as the meet of these atoms, that is, 2 Ù 3 Ù 5:
2 3 5 (ATOMS, RANK=1) \ | / \ | / \ | / \ | / \ | / \ | / \ | / 1 (min=GCD(2,3,5), RANK=0)- Now, we can consider the least common multiples of the atoms, taken in pairs, namely, LCM(2, 3) = 6, LCM(2,6) = 10, LCM(3,5) = 15, and graph these into the Hasse diagram:
6 10 15 (RANK = 2) |\ / \ /| | \ / \ / | | \ / \ / | | \ / \ / | | / \ | | / \ / \ | | / \ / \ | | / \ / \ | 2 3 5 (ATOMS, RANK = 1) \ | / \ | / \ | / \ | / \ | / \ | / \ | / 1 (min=GCD(2,3,5), RANK = 0)- Finally, we consider the least common multiple of 6, 10, 15, namely, LCM(6, 10, 15) = 30, and put in the Hasse graph of these as their join, namely, 6 Ù 10 Ú 15:
30 (max = LCM(2,3,5), RANK = 3, FILE = 1) /\ / \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ 6 10 15 (LCM's of ATOMS, RANK = 2, FILE = 3) |\ / \ /| | \ / \ / | | \ / \ / | | \ / \ / | | / \ | | / \ / \ | | / \ / \ | | / \ / \ | 2 3 5 (ATOMS, RANK = 1, FILE = 3) \ | / \ | / \ | / \ | / \ | / \ | / \ | / 1 (min = GCD(2,3,5), RANK = 0, FILE = 1)
For simplicity and to save room in this file, I consider the simplest form of an o-lattice, the factor lattice on 12 = 2*2*3. It is the multiplicity of the prime factor 2 that creates the difference between a o-lattice and a t-lattice. For, a t-lattice (as with a t-set) respects only type (a.k.a. kind), whereas an o-lattice (as with my o-sets) respects both type and order (a.k.a. degree). And this introduction of order-degree graphs in a distinguishable way:2 @ 2 Ú 3 = 12 (max) /\ / \ / \ 2 @ 2 = 4 2 Ú 3 = 6 | /| | / | gopain -> | / | <- bierarch | / | | / | |/ | 2 3 (atoms) \ / \ / \/ 1 (min)The sublattice consisting of {1, 2, 3, 6} is bierarch (partial ordering), as in a t-lattice (which you see indicated in the diagram by an arrow). But the sublattice consisting of {2, 4, 12} doesn't fit this pattern. It is a chain-pattern (simple ordering). And notice that {2, 4} is a geometric progression, that is, the chain is homologous to a geometric progression. Hence, as noted above, I created a new label for this, "gopain", or elliptically, g(e)o(metric)p(rogression ch)ain. And you see this indicated in the diagram by annother arrow.Also, note that I had to create a new operator for its "chaining". The two main lattice operators of join, meet (homologues of arithmetical LCM, GCD) are binary operators, whereas a unary operation is now called for. For this, I created the label. "aug" (from "augment"), and denoted it as @, hence, I can insert in the diagram 2 @ 2 = 4. Furthermore, since 12 = 2 * 2 * 3 is doubly two-ed, singly 3-ed, its operational notation is a mixture of @, Ú, namely, 2 @ 2 Ú 3 = 12.
So the factor lattice provides the model. Its bierach patterm fits an instance of t-math (lattices, sets, logic, probabilities); its gopain pattern fits the peculiar aspect of any o-math (....). For, example, I am unique in setting forth an o-probability, imposing degrees of event, along with type, and I consistently parse out the subprobabilities by a gopain in the lattice representation, evaluated by resort to a geometric progression of "atomic" probabilities.Elsewhere (realev.html), I note the importance of this -- not merely, theoretically, but in application. Probabilities can into evolutionary theory with Mendelian genetics. But it has remained t-probability -- because that's all they know, without examining my work. Yet, evolution of type or kind is mostly or entirely inferred, whereas instances of evolution of order or degree have actually been observed. (Example: darkening of wings of moths "living" on tree trunks in Gt. Britain that have become darkened by coal-pollution.) So, inferred evolution has been mathematized, but observed evolution has not. Do you care enough to change this?