Properties
A subset S of a partially ordered set P may fail to have any bounds or may have many different upper and lower bounds. By transitivity, any element greater than or equal to an upper bound of S is again an upper bound of S, and any element less than or equal to any lower bound of S is again a lower bound of S. This leads to the consideration of least upper bounds (or suprema) and greatest lower bounds (or infima).
The bounds of a subset S of a partially ordered set K may or may not be elements of S itself. If S contains an upper bound then that upper bound is unique and is called the greatest element of S. The greatest element of S (if it exists) is also the least upper bound of S. A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds. This observation leads to the definition of Dedekind cuts.
The empty subset ∅ of a partially ordered set K is conventionally considered to be both bounded from above and bounded from below with every element of P being both an upper and lower bound of ∅.
Read more about this topic: Upper And Lower Bounds
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)