In other words, a partial order is an antisymmetric preorder. A set with a partial order is called a partially ordered set. The term ordered set is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. For a, b distinct elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. If every two elements of a poset are comparable, the poset is called a totally ordered set or chain. A poset in which every two elements are incomparable is called an antichain.
Examples
Standard examples of posets arising in mathematics include:
The real numbers ordered by the standard less-than-or-equal relation ≤.
The set of subsets of a given set ordered by inclusion
The set of subspaces of a vector space ordered by inclusion.
For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, if and only if for all n in ℕ.
For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f ≤ g for all x in X.
There are several notions of "greatest" and "least" element in a poset P, notably:
Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g. An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest or least element.
Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a in P such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a < m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.
Upper and lower bounds: For a subsetA of P, an element x in P is an upper bound of A if a ≤ x, for each element a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is a lower bound of A if a ≥ x, for each element a in A. A greatest element ofP is an upper bound of P itself, and a least element is a lower bound of P.
For example, consider the natural numbers, ordered by divisibility: 1 is a least element, as it divides all other elements, but this set does not have a greatest element nor does it have any maximal elements: any g divides 2g, so 2g is greater thang and g cannot be maximal. If instead we consider only the natural numbers that are greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element. In this poset, 60 is an upper bound of and 2 is a lower bound of.
A poset is a lattice if it satisfies the following two axioms. ;Existence of binary joins: ;Existence of binary meets: The join and meet of a and b are denoted by and, respectively. This definition makes and binary operations. The first axiom says that L is a join-semilattice; the second says that L is a meet-semilattice. Both operations are monotone with respect to the order: a1 ≤ a2 and b1 ≤ b2 implies that a1 b1 ≤ a2 b2 and a1b1 ≤ a2b2. It follows by an induction argument that every non-empty finite subset of a lattice has a join and a meet. With additional assumptions, further conclusions may be possible; see Completeness for more discussion of this subject. A bounded lattice has a greatest and least element, denoted 1 and 0 by convention. Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every non-empty finite lattice is bounded, by taking the join of all elements, denoted by where. A poset is a bounded lattice if and only if every finite set of elements has a join and a meet. Here, the join of an empty set of elements is defined to be the least element , and the meet of the empty set is defined to be the greatest element. This convention is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L, and hold. Taking B to be the empty set, and which is consistent with the fact that.
Ordered algebraic structure
The poset can be a partially ordered algebraic structure. In algebra, an ordered semigroup is a semigroup together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid. Partially ordered vector spaces and vector lattices are important in optimization with multiple objectives.
Problems of ordinal optimization arise in many disciplines. Computer scientists study selection algorithms, which are simpler than sorting algorithms. Statistical decision theory studies "selection problems" that require the identification of a "best" subpopulation or of identifying a "near best" subpopulation.
Applications
Since the 1960s, the field of ordinal optimization has expanded in theory and in applications. In particular, antimatroids and the "max-plus algebra" have found application in network analysis and queuing theory, particularly in queuing networks and discrete-event systems.
