Properties
A strict weak ordering has the following properties. For all x and y in S,
- For all x, it is not the case that x < x (irreflexivity).
- For all x, y, if x < y then it is not the case that y < x (asymmetric).
- For all x, y, and z, if x < y and y < z then x < z (transitivity).
- For all x, y, and z, if x is incomparable with y, and y is incomparable with z, then x is incomparable with z (transitivity of incomparability).
This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.
Transitivity of incomparability (together with transitivity) can also be stated in the following forms:
- If x < y, then for all z, either x < z or z < y or both.
Or:
- If x is incomparable with y, then for all z≠x, z≠y, either (x < z and y < z) or (z < x and z < y) or (z is incomparable with x and z is incomparable with y).
Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.
Read more about this topic: Strict Weak Ordering
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)