GM-RKB: Ordered Set

AKA: Ordered, Total Strict Ordered Set.
Context:
- It can be:
  - a Finite Ordered Set (Ordinal Set).
  - an Non-Finite Ordered Set.
- It can be a Permutation.
- It can be Denote by {a < b < ... < c < d }
- It can represent a Data Type.
Example(s):
- {a < b < c < d }.
- (bad < medium < good)
- {Large > Medium > Small}
- (first < second < third)
- {First < Second < Third < ... < Infinity}
- {Soft < Medium < Hard}.
- {First < Second < Third}, a Cardinal Set.
- a Formal Number Sequence.
Counter-Example(s):
- {True, False}, an Unordered Set.
- (c, e, d, e), a Sequence with repeated Members (an Ordered Multiset)
- (soft ≤ medium ≤ medium ≤ hard), a Sequence.
See: Unordered Set, Ordinal, Scale, Interval Scale, Categorical Set, Monotone Function.

References

http://en.wiktionary.org/wiki/ordinal_scale
- Noun
  - 1. A scale whose values can be compared; formally, a scale whose set of values is totally ordered.
- Usage notes: An equivalent definition: A scale that defines a total preorder on the set of measured objects.
(Wikipedia, 2009) http://en.wikipedia.org/wiki/Ordinal_scale
- An ordinal scale defines a total preorder of objects; the scale values themselves have a total order; names may be used like "bad", "medium", "good"; if numbers are used they are only relevant up to strictly monotonically increasing transformations (order isomorphism).
(Wikipedia, 2009) http://en.wikipedia.org/wiki/Level_of_measurement#Ordinal_scale
- In this scale type, the numbers assigned to objects or events represent the rank order (1st, 2nd, 3rd etc.) of the entities assessed. An example of ordinal measurement is the results of a horse race, which say only which horses arrived first, second, third, etc. but include no information about times. Another is the Mohs scale of mineral hardness, which characterizes the hardness of various minerals through the ability of a harder material to scratch a softer one, saying nothing about the actual hardness of any of them.

(Li & Link 2006) => Ling Li, and Hsuan-Tien Lin. (2006). "Ordinal Regression by Extended Binary Classification. In: Advances in Neural Information Processing Systems 19 (NIPS 2007).
- Notes; It defines a Ordinal Label/Rank as: "ordinal label (i.e., rank) y∈Y = {1, 2, . . . ,K}."

(Kohavi & Provost, 1998) => Ron Kohavi, and Foster Provost. (1998). "Glossary of Terms." In: Machine Leanring 30(2-3).
- Ordinal: A finite number of discrete values. The type nominal denotes that there is no ordering between the values, such as last names and colors. The type ordinal denotes that there is an ordering, such as in an attribute taking on the values low, medium, or high.