Related Structures
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.
A group with a compatible total order is a totally ordered group.
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data results in a separation relation.
Read more about this topic: Total Order
Famous quotes containing the words related and/or structures:
“A parent who from his own childhood experience is convinced of the value of fairy tales will have no difficulty in answering his child’s questions; but an adult who thinks these tales are only a bunch of lies had better not try telling them; he won’t be able to related them in a way which would enrich the child’s life.”
—Bruno Bettelheim (20th century)
“The American who has been confined, in his own country, to the sight of buildings designed after foreign models, is surprised on entering York Minster or St. Peter’s at Rome, by the feeling that these structures are imitations also,—faint copies of an invisible archetype.”
—Ralph Waldo Emerson (1803–1882)