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Partially ordered set  ( Order Theory )

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In , especially , a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The word "partial" in the names "partial order" or "partially ordered set" is used as an indication that not every pair of elements need be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize , in which every pair is comparable.

To be a partial order, a binary relation must be reflexive (each element is comparable to itself), antisymmetric (no two different elements precede each other), and transitive (the start of a chain of precedence relations must precede the end of the chain).

One familiar example of a partially ordered set is a collection of people ordered by descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendent of the other.

A poset can be visualized through its , which depicts the ordering relation.

(1989). 9780471838173, John Wiley & Sons. .

Formal definition
A (non-strict) partial order
(2018). 9781848002012, Springer. .
is a ≤ over a set P satisfying particular axioms which are discussed below. When ab, we say that a is related to b. (This does not imply that b is also related to a, because the relation need not be symmetric.)

The axioms for a non-strict partial order state that the relation ≤ is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy:

1. aa (reflexivity: every element is related to itself).
2. if ab and ba, then a = b (antisymmetry: two distinct elements cannot be related in both directions).
3. if ab and bc, then ac (transitivity: if a first element is related to a second element, and, in turn, that element is related to a third element, then the first element is related to the third element).

In other words, a partial order is an antisymmetric .

A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used, as long as it is clear from the context that no other kind of order is meant. In particular, can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.

For a, b, elements of a partially ordered set P, if ab or ba, then a and b are . Otherwise they are incomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partial order under which every pair of elements is comparable is called a total order or linear order; a totally ordered set is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no two distinct elements are comparable is called an (e.g. the set of singletons in the top-right figure). An element a is said to be covered by another element b, written a<: b, if a is strictly less than b and no third element c fits between them; formally: if both ab and ab are true, and acb is false for each c with acb. A more concise definition will be given below using the strict order corresponding to "≤". For example, {x} is covered by {x,z} in the top-right figure, but not by {x,y,z}.

Examples
Standard examples of posets arising in mathematics include:

• The ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).
• The set of of a given set (its ) ordered by (see the figure on top-right). Similarly, the set of ordered by , and the set of strings ordered by .
• The set of equipped with the relation of divisibility.
• The vertex set of a directed acyclic graph ordered by .
• The set of of a ordered by inclusion.
• For a partially ordered set P, the containing all of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, if and only if for all n in ℕ, i.e. a componentwise order.
• For a set X and a partially ordered set P, the containing all functions from X to P, where fg if and only if f( x) ≤ g( x) for all x in X.
• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...
• The set of events in special relativity, where for two events X and Y, X ≤ Y if and only if Y is in the future of X. An event Y can only be causally affected by X if X ≤ Y.

Extrema
There are several notions of "greatest" and "least" element in a poset P, notably:
• 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.
• 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 subset A 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 of P is an upper bound of P itself, and a least element is a lower bound of P.

For example, consider the , ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, which is a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not even have any maximal elements, since any g divides for instance 2 g, which is distinct from it, so g is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

Orders on the Cartesian product of partially ordered sets
by (3,3) and covering (3,3) are highlighted in green and red, respectively.]]
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see figures):
• the lexicographical order:   ( a, b) ≤ ( c, d) if a < c or ( a = c and bd);
• the :   ( a, b) ≤ ( c, d) if ac and bd;
• the reflexive closure of the direct product of the corresponding strict orders:   ( a, b) ≤ ( c, d) if ( a < c and b < d) or ( a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets.

Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.

Sums of partially ordered sets
Another way to combine two posets is the ordinal sum (or linear sum
(2018). 9780521784511, Cambridge University Press. .
), Z = XY, defined on the union of the underlying sets X and Y by the order a Z b if and only if:
• a, bX with a X b, or
• a, bY with a Y b, or
• aX and bY.

If two posets are , then so is their ordinal sum.

(1974). 9781475716450, Springer.
The ordinal sum operation is one of two operations used to form series-parallel partial orders, and in this context is called series composition. The other operation used to form these orders, the disjoint union of two partially ordered sets (with no order relation between elements of one set and elements of the other set) is called in this context parallel composition.

Strict and non-strict partial orders
In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In these contexts, a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric, i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity),
• if a < b and b < c then a < c (transitivity), and
• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".).

Strict and non-strict partial orders are closely related. A non-strict partial order may be converted to a strict partial order by removing all relationships of the form aa. Conversely, a strict partial order may be converted to a non-strict partial order by adjoining all relationships of that form. Thus, if "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if ab and ab
Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closure given by:

ab if a < b or a = b.
This is the reason for using the notation "≤".

Using the strict order "<", the relation " a is covered by b" can be equivalently rephrased as " a< b, but not a< c< b for any c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

Inverse and order dual
The or converse ≥ of a partial order relation ≤ satisfies xy if and only if yx. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.

Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.

In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The , the , the , and the are all totally ordered by their algebraic (signed) magnitude whereas the are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+ i y < u +iv if and only if x < u or ( x = u and y < v), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100 i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.

Mappings between partially ordered sets
Given two partially ordered sets ( S,≤) and ( T,≤), a function f: ST is called , or monotone, or isotone, if for all x and y in S, xy implies f( x) ≤ f( y). If ( U,≤) is also a partially ordered set, and both f: ST and g: TU are order-preserving, their composition ( gf): SU is order-preserving, too. A function f: ST is called order-reflecting if for all x and y in S, f( x) ≤ f( y) implies xy. If f is both order-preserving and order-reflecting, then it is called an of ( S,≤) into ( T,≤). In the latter case, f is necessarily , since f( x) = f( y) implies xy and yx. If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding f: ST is , it is called an order isomorphism, and the partial orders ( S,≤) and ( T,≤) are said to be isomorphic. Isomorphic orders have structurally similar (cf. right picture). It can be shown that if order-preserving maps f: ST and g: TS exist such that gf and fg yields the identity function on S and T, respectively, then S and T are order-isomorphic.
(2018). 9780521784511, Cambridge University Press.
.

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its . It is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neither injective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Taking instead each number to the set of its divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number to the set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and its isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices, see "Birkhoff's representation theorem".

Number of partial orders
Sequence in OEIS gives the number of partial orders on a set of n labeled elements:

The number of strict partial orders is the same as that of partial orders.

If the count is made only isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, … is obtained.

Linear extension
A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and y of X, whenever xy, it is also the case that x ≤*  y. A is an extension that is also a linear (i.e., total) order. Every partial order can be extended to a total order (order-extension principle).
(2018). 9780486466248, Dover Publications.

In , algorithms for finding linear extensions of partial orders (represented as the orders of directed acyclic graphs) are called topological sorting.

In category theory
Every poset (and every ) may be considered as a category in which every hom-set has at most one element. More explicitly, let hom( x, y) = {( x, y)} if xy (and otherwise the empty set) and ( y, z)∘( x, y) = ( x, z). Such categories are sometimes called posetal.

Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an , and the largest element, if it exists, is a . Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

Partial orders in topological spaces
If P is a partially ordered set that has also been given the structure of a topological space, then it is customary to assume that is a closed subset of the topological $P\times P$. Under this assumption partial order relations are well behaved at limits in the sense that if $a_i\to a$, $b_i\to b$ and a i ≤  b i for all i, then a ≤  b.

Interval
For ab, the closed interval is the set of elements x satisfying axb (i.e. ax and xb). It contains at least the elements a and b.

Using the corresponding strict relation "<", the is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b. For example, the open interval on the integers is empty since there are no integers i such that 1 < i < 2.

Sometimes the definitions are extended to allow a > b, in which case the interval is empty.

The half-open intervals and are defined similarly.

A poset is locally finite if every interval is finite. For example, the are locally finite under their natural ordering. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1). Using the interval notation, the property " a is covered by b" can be rephrased equivalently as a, b = { a, b}.

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the .

• , a formalization of orderings on a set that allows more general families of orderings than posets
• comparability graph
• complete partial order
• incidence algebra
• lattice
• locally finite poset
• Möbius function on posets
• , a kind of topological space that can be defined from any poset
• – continuity of a function between two partial orders.
• stochastic dominance
• strict weak ordering – strict partial order "<" in which the relation is transitive.
• Zorn's lemma

Notes

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monotonic but nonhomomorphic map between lattices.gif (2018-04-05)

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