Interval
For a ≤ b, the closed interval is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains at least the elements a and b.
Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers is empty since there are no integers i such that 1 < i < 2.
Sometimes the definitions are extended to allow a > b, in which case the interval is empty.
The half-open intervals are defined similarly.
A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural ordering.
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.
Read more about this topic: Partially Ordered Set
Famous quotes containing the word interval:
“The yearning for an afterlife is the opposite of selfish: it is love and praise for the world that we are privileged, in this complex interval of light, to witness and experience.”
—John Updike (b. 1932)
“[I have] been in love with one princess or another almost all my life, and I hope I shall go on so, till I die, being firmly persuaded, that if ever I do a mean action, it must be in some interval betwixt one passion and another.”
—Laurence Sterne (1713–1768)
“I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater interval of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.”
—Henry David Thoreau (1817–1862)