Total Strict Ordered Set

Context:
- It can range from being a Finite Ordered Set (Ordinal Set) to being a Non-Finite Ordered Set.
- It can be a Permutation.
- It can be Denote by {a < b < ... < c < d }
- It can be represented by an Ordered Set Data Structure.
- 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 Ranked List, such as a TOP500 List.
- a Formal Number Sequence.
- …
Counter-Example(s):
- an Unordered Set, such as {True, False}.
- an Ordered Multiset, such as (c, e, d, e).
- a Sequence, such as (soft ≤ medium ≤ medium ≤ hard).
- a Categorical Set.
See: Ordinal, Level of Measurement, Interval Scale, Monotone Function.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Level_of_measurement#Ordinal_scale Retrieved:2015-6-1.
- The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted, but still does not allow for relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as 'sick' vs. 'healthy' when measuring health, 'guilty' vs. 'innocent' when making judgments in courts, 'wrong/false' vs. 'right/true' when measuring truth value, and, on the other hand, non-dichotomous data consisting of a spectrum of values, such as 'completely agree', 'mostly agree', 'mostly disagree', 'completely disagree' when measuring opinion.

http://en.wiktionary.org/wiki/ordinal_scale
- 1. A scale whose values can be compared; formally, a scale whose set of values is totally ordered.

(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 2006).
- Notes; It defines an 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.