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
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