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:
“Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained—a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas.”
—Chauncey Wright (1830–1875)
“A proper secrecy is the only mystery of able men; mystery is the only secrecy of weak and cunning ones.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“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)