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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets ), orderings in which some pairs are comparable and others might not be
A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on natural numbers , integers , rational numbers and reals are all orders in the above sense.
Order topology, a topology of total order for totally ordered sets; Ordinal numbers, numbers assigned to sets based on their set-theoretic order; Partial order, often called just "order" in order theory texts, a transitive antisymmetric relation
Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties: x and y are incomparable in P, for every z in S, if z < x then z < y, and; for every z in S, if y < z then x < z. If (x, y) is a critical pair, then the binary relation obtained from P by adding the single relationship x ≤ y is also a partial
Every well-ordered set is order-equivalent to exactly one ordinal number, by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.