In mathematics, especially order theory, a strict weak ordering is a binary relation < on a set S that is a strict partial order (a transitive relation that is irreflexive, or equivalently, that is asymmetric) in which the relation "neither a < b nor b < a" is transitive.
The equivalence classes of this "incomparability relation" partition the elements of S, and are totally ordered by <. Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x < y if and only if there exists sets A and B in the partition with x in A, y in B, and A < B in the total order.
As a non-example, consider the partial order in the set {a, b, c} defined by the relationship b < c. The pairs a,b and a,c are incomparable but b and c are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering.
Read more about Strict Weak Ordering: Properties, Total Preorders, Representing Weak Orderings By Functions, The Number of Total Preorders, Strict Total Order
Famous quotes containing the words strict, weak and/or ordering:
“To safeguard one’s health at the cost of too strict a diet is a tiresome illness indeed.”
—François, Duc De La Rochefoucauld (1613–1680)
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)
“Our goal as a parent is to give life to our children’s learning—to instruct, to teach, to help them develop self-discipline—an ordering of the self from the inside, not imposition from the outside. Any technique that does not give life to a child’s learning and leave a child’s dignity intact cannot be called discipline—it is punishment, no matter what language it is clothed in.”
—Barbara Coloroso (20th century)