{"history":[],"article":{"id":"06836020-c510-4904-b73f-fc744ea7dcf8","title":"Partially ordered set","slug":"partially-ordered-set","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), especially [order theory](https://bliptext.com/articles/order-theory), a **partial order** on a [set](https://bliptext.com/articles/set-mathematics) is an arrangement such that, for certain pairs of elements, one precedes the other.\nThe word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other.\nPartial orders thus generalize [total orders](https://bliptext.com/articles/total-order), in which every pair is comparable.\nFormally, a partial order is a [homogeneous binary relation](https://bliptext.com/articles/homogeneous-binary-relation) that is [reflexive](https://bliptext.com/articles/reflexive-relation), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), and [transitive](https://bliptext.com/articles/transitive-relation).\nA **partially ordered set** (**poset** for short) is an ordered pair P=(X,\\leq) consisting of a set X (called the ground set of P) and a partial order \\leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a **poset**.\n# [Fig. 1 The Hasse diagram of the set of all subsets of a three-element set ordered by inclusion. Sets connected by an upward path, like \\emptyset and \\{x,y\\}, are comparable, while e.g. \\{x\\} and \\{y\\} are not. | upload.wikimedia.org/wikipedia/commons/e/ea/Hasse///diagram///of///powerset///of///3.svg]\n\n# Partial order relations\nThe term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders.\nHowever some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders.\nStrict and non-strict partial orders can be put into a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence), so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.\n\n## Partial orders\n****A**** **************r****e****f****l****e****x****i****v****e**************,**** **************w****e****a****k**************,**** ****o****r**** ****,**** ****c****o****m****m****o****n****l****y**** ****r****e****f****e****r****r****e****d**** ****t****o**** ****s****i****m****p****l****y**** ****a****s**** ****a**** ****p****a****r****t****i****a****l**** ****o****r****d****e****r****,**** ****i****s**** ****a**** ****h****o****m****o****g****e****n****e****o****u****s**** ****r****e****l****a****t****i****o****n**** ****≤**** ****o****n**** ****a**** ****s****e****t**** ****P**** ****t****h****a****t**** ****i****s**** **************r****e****f****l****e****x****i****v****e**************,**** ****a****n****t****i****s****y****m****m****e****t****r****i****c****,**** ****a****n****d**** ****t****r****a****n****s****i****t****i****v****e****.****\nThat is, for all it must satisfy:\nA non-strict partial order is also known as an antisymmetric [preorder](https://bliptext.com/articles/preorder).\n\n## Strict partial orders\n****A****n**** **************i****r****r****e****f****l****e****x****i****v****e**************,**** **************s****t****r****o****n****g**************,**** ****o****r**** **** ****i****s**** ****a**** ****h****o****m****o****g****e****n****e****o****u****s**** ****r****e****l****a****t****i****o****n**** ****<**** ****o****n**** ****a**** ****s****e****t**** ****P**** ****t****h****a****t**** ****i****s**** ****t****r****a****n****s****i****t****i****v****e****,**** **************i****r****r****e****f****l****e****x****i****v****e**************,**** ****a****n****d**** ****a****s****y****m****m****e****t****r****i****c****;**** ****t****h****a****t**** ****i****s****,**** ****i****t**** ****s****a****t****i****s****f****i****e****s**** ****t****h****e**** ****f****o****l****l****o****w****i****n****g**** ****c****o****n****d****i****t****i****o****n****s**** ****f****o****r**** ****a****l****l****\nA transitive relation is asymmetric if and only if it is irreflexive.\nSo the definition is the same if it omits either irreflexivity or asymmetry (but not both).\nA strict partial order is also known as an asymmetric [strict preorder](https://bliptext.com/articles/strict-preorder).\n\n## Correspondence of strict and non-strict partial order relations\nStrict and non-strict partial orders on a set P are closely related.\nA non-strict partial order \\leq may be converted to a strict partial order by removing all relationships of the form a \\leq a; that is, the strict partial order is the set where is the [identity relation](https://bliptext.com/articles/identity-relation) on P \\times P and denotes [set subtraction](https://bliptext.com/articles/set-subtraction).\nConversely, a strict partial order < on P may be converted to a non-strict partial order by adjoining all relationships of that form; that is, is a non-strict partial order.\nThus, if \\leq is a non-strict partial order, then the corresponding strict partial order < is the [irreflexive kernel](https://bliptext.com/articles/irreflexive-kernel) given by\nConversely, if < is a strict partial order, then the corresponding non-strict partial order \\leq is the [reflexive closure](https://bliptext.com/articles/reflexive-closure) given by:\n# [Fig. 2 Commutative diagram about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (cls), irreflexive kernel (ker), and converse relation (cnv). Each relation is depicted by its logical matrix for the poset whose Hasse diagram is depicted in the center. For example so row 3, column 4 of the bottom left matrix is empty. | upload.wikimedia.org/wikipedia/commons/4/42/PartialOrders///redundencies///svg.svg]\n\n## Dual orders\nThe dual (or opposite) of a partial order relation R is defined by letting be the [converse relation](https://bliptext.com/articles/converse-relation) of R, i.e. if and only if y R x. The dual of a non-strict partial order is a non-strict partial order, and the dual of a strict partial order is a strict partial order.\nThe dual of a dual of a relation is the original relation.\n\n# Notation\nGiven a set P and a partial order relation, typically the non-strict partial order \\leq, we may uniquely extend our notation to define four partial order relations \\leq, <, \\geq, and >, where \\leq is a non-strict partial order relation on P, < is the associated strict partial order relation on P (the [irreflexive kernel](https://bliptext.com/articles/irreflexive-kernel) of \\leq), \\geq is the dual of \\leq, and > is the dual of <.\nStrictly speaking, the term partially ordered set refers to a set with all of these relations defined appropriately.\nBut practically, one need only consider a single relation, (P,\\leq) or (P,<), or, in rare instances, the non-strict and strict relations together, (P,\\leq,<).\nThe term ordered set is sometimes used as a shorthand for partially ordered set, as long as it is clear from the context that no other kind of order is meant.\nIn particular, [totally ordered sets](https://bliptext.com/articles/total-order) can also be referred to as \"ordered sets\", especially in areas where these structures are more common than posets.\nSome authors use different symbols than \\leq such as \\sqsubseteq or \\preceq to distinguish partial orders from total orders.\nWhen referring to partial orders, \\leq should not be taken as the [complement](https://bliptext.com/articles/complementary-relation) of >.\nThe relation > is the converse of the irreflexive kernel of \\leq, which is always a subset of the complement of \\leq, but > is equal to the complement of \\leq [if, and only if](https://bliptext.com/articles/if-and-only-if), \\leq is a total order.\n\n# Alternative definitions\nAnother way of defining a partial order, found in [computer science](https://bliptext.com/articles/computer-science), is via a notion of [comparison](https://bliptext.com/articles/comparability).\nSpecifically, given as defined previously, it can be observed that two elements x and y may stand in any of four [mutually exclusive](https://bliptext.com/articles/mutually-exclusive) relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable.\nThis can be represented by a function that returns one of four codes when given two elements.\nThis definition is equivalent to a partial order on a [setoid](https://bliptext.com/articles/setoid), where equality is taken to be a defined [equivalence relation](https://bliptext.com/articles/equivalence-relation) rather than set equality.\nWallis defines a more general notion of a partial order relation as any [homogeneous relation](https://bliptext.com/articles/homogeneous-relation) that is [transitive](https://bliptext.com/articles/transitive-relation) and [antisymmetric](https://bliptext.com/articles/antisymmetric-relation).\nThis includes both reflexive and irreflexive partial orders as subtypes.\nA finite poset can be visualized through its [Hasse diagram](https://bliptext.com/articles/hasse-diagram).\nSpecifically, taking a strict partial order relation (P,<), a [directed acyclic graph](https://bliptext.com/articles/directed-acyclic-graph) (DAG) may be constructed by taking each element of P to be a node and each element of < to be an edge.\nThe [transitive reduction](https://bliptext.com/articles/transitive-reduction) of this DAG is then the Hasse diagram.\nSimilarly this process can be reversed to construct strict partial orders from certain DAGs.\nIn contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.\n\n# Examples\nStandard examples of posets arising in mathematics include:\nOne familiar example of a partially ordered set is a collection of people ordered by [genealogical](https://bliptext.com/articles/genealogy) descendancy.\nSome pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.\n# [Fig. 3 Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4 | upload.wikimedia.org/wikipedia/commons/f/f3/Division///relation///4.svg]\n\n## Orders on the Cartesian product of partially ordered sets\nIn order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the [Cartesian product](https://bliptext.com/articles/cartesian-product) of two partially ordered sets are (see Fig. 4):\nAll three can similarly be defined for the Cartesian product of more than two sets.\nApplied to [ordered vector spaces](https://bliptext.com/articles/ordered-vector-space) over the same [field](https://bliptext.com/articles/field-mathematics), the result is in each case also an ordered vector space.\nSee also [orders on the Cartesian product of totally ordered sets](https://bliptext.com/articles/total-order).\n\n## Sums of partially ordered sets\nAnother way to combine two (disjoint) posets is the **ordinal sum** (or **linear sum**), Z = X ⊕ Y, defined on the union of the underlying sets X and Y by the order a ≤Z b if and only if:\nIf two posets are [well-ordered](https://bliptext.com/articles/well-ordered), then so is their ordinal sum.\n[Series-parallel partial orders](https://bliptext.com/articles/series-parallel-partial-order) are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition.\nParallel composition is the [disjoint union](https://bliptext.com/articles/disjoint-union) of two partially ordered sets, with no order relation between elements of one set and elements of the other set.\n\n# Derived notions\nThe examples use the poset consisting of the [set of all subsets](https://bliptext.com/articles/power-set) of a three-element set ordered by set inclusion (see Fig. 1).\n\n## Extrema\nThere are several notions of \"greatest\" and \"least\" element in a poset P, notably:\nAs another example, consider the positive [integers](https://bliptext.com/articles/integer), ordered by divisibility: 1 is a least element, as it [divides](https://bliptext.com/articles/divisor) all other elements; on the other hand this poset does not have a greatest element.\nThis partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal.\nIf 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 [prime number](https://bliptext.com/articles/prime-number) is a minimal element for it.\nIn this poset, 60 is an upper bound (though not a least upper bound) of the subset 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.\nIf the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig. 6).\n# [Fig. 5 The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all elements, and the bottom row are all elements, but there is no and no element. | upload.wikimedia.org/wikipedia/commons/9/9e/Hasse///diagram///of///powerset///of///3///no///greatest///or///least.svg]\n# [Fig. 6 Nonnegative integers, ordered by divisibility | upload.wikimedia.org/wikipedia/commons/e/e6/Infinite///lattice///of///divisors.svg]\n\n# Mappings between partially ordered sets\nGiven two partially ordered sets\n(S, ≤)\nand\n(T, ≼)\n, a function f : S \\to T is called **[order-preserving](https://bliptext.com/articles/order-preserving)**, or **[monotone](https://bliptext.com/articles/monotonic-function)**, or **isotone**, if for all x, y \\in S, x \\leq y implies\nf(x) ≼ f(y)\n. If\n(U, ≲)\nis also a partially ordered set, and both f : S \\to T and g : T \\to U are order-preserving, their [composition](https://bliptext.com/articles/function-composition) is order-preserving, too.\nA function f : S \\to T is called **order-reflecting** if for all x, y \\in S,\nf(x) ≼ f(y)\nimplies x \\leq y. If f is both order-preserving and order-reflecting, then it is called an **[order-embedding](https://bliptext.com/articles/order-embedding)** of\n(S, ≤)\ninto\n(T, ≼)\n. In the latter case, f is necessarily [injective](https://bliptext.com/articles/injective), since f(x) = f(y) implies and in turn x = y according to the antisymmetry of \\leq.\nIf an order-embedding between two posets S and T exists, one says that S can be **embedded** into T.\nIf an order-embedding f : S \\to T is [bijective](https://bliptext.com/articles/bijective), it is called an **[order isomorphism](https://bliptext.com/articles/order-isomorphism)**, and the partial orders\n(S, ≤)\nand\n(T, ≼)\nare said to be **isomorphic**.\nIsomorphic orders have structurally similar [Hasse diagrams](https://bliptext.com/articles/hasse-diagram) (see Fig. 7a).\nIt can be shown that if order-preserving maps f : S \\to T and g : T \\to U exist such that g \\circ f and f \\circ g yields the [identity function](https://bliptext.com/articles/identity-function) on S and T, respectively, then S and T are order-isomorphic.\nFor example, a mapping from the set of natural numbers (ordered by divisibility) to the [power set](https://bliptext.com/articles/power-set) of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its [prime divisors](https://bliptext.com/articles/prime-divisor).\nIt 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 12 does not divide 6).\nTaking instead each number to the set of its [prime power](https://bliptext.com/articles/prime-power) divisors defines a map that is order-preserving, order-reflecting, and hence an order-embedding.\nIt is not an order-isomorphism (since it, for instance, does not map any number to the set \\{4\\}), but it can be made one by [restricting its codomain](https://bliptext.com/articles/injective-function) to g(\\N).\nFig. 7b 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](https://bliptext.com/articles/distributive-lattice); see [Birkhoff's representation theorem](https://bliptext.com/articles/birkhoff-s-representation-theorem).\n\n# Number of partial orders\nSequence [ A001035] in [OEIS](https://bliptext.com/articles/on-line-encyclopedia-of-integer-sequences) gives the number of partial orders on a set of n labeled elements:\nThe number of strict partial orders is the same as that of partial orders.\nIf the count is made only [up to](https://bliptext.com/articles/up-to) isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.\n\n# Subposets\nA poset is called a **subposet** of another poset P=(X, \\leq) provided that X^* is a [subset](https://bliptext.com/articles/subset) of X and \\leq^* is a [subset](https://bliptext.com/articles/subset) of \\leq.\nThe latter condition is equivalent to the requirement that for any x and y in X^* (and thus also in X), if x \\leq^* y then x \\leq y.\nIf P^* is a subposet of P and furthermore, for all x and y in X^*, whenever x \\leq y we also have x \\leq^* y, then we call P^* the subposet of P **induced** by X^*, and write.\n\n# Linear extension\nA partial order \\leq^* on a set X is called an **extension** of another partial order \\leq on X provided that for all elements x, y \\in X, whenever x \\leq y, it is also the case that x \\leq^* y. A linear **extension** is an **extension** that is also a linear (that is, total) order.\nAs a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.\nEvery partial order can be extended to a total order ([order-extension principle](https://bliptext.com/articles/order-extension-principle)).\nIn [computer science](https://bliptext.com/articles/computer-science), algorithms for finding linear extensions of partial orders (represented as the [reachability](https://bliptext.com/articles/reachability) orders of [directed acyclic graphs](https://bliptext.com/articles/directed-acyclic-graph)) are called [topological sorting](https://bliptext.com/articles/topological-sorting).\n\n# In category theory\nEvery poset (and every [preordered set](https://bliptext.com/articles/preorder)) may be considered as a [category](https://bliptext.com/articles/category-mathematics) where, for objects x and y, there is at most one [morphism](https://bliptext.com/articles/morphism) from x to y. More explicitly, let hom(x, y) = if x ≤ y (and otherwise the [empty set](https://bliptext.com/articles/empty-set)) and Such categories are sometimes called [posetal](https://bliptext.com/articles/posetal-category).\nPosets are [equivalent](https://bliptext.com/articles/equivalence-of-categories) to one another if and only if they are [isomorphic](https://bliptext.com/articles/isomorphism-of-categories).\nIn a poset, the smallest element, if it exists, is an [initial object](https://bliptext.com/articles/initial-object), and the largest element, if it exists, is a [terminal object](https://bliptext.com/articles/terminal-object).\nAlso, every preordered set is equivalent to a poset.\nFinally, every subcategory of a poset is [isomorphism-closed](https://bliptext.com/articles/isomorphism-closed).\n\n# Partial orders in topological spaces\nIf P is a partially ordered set that has also been given the structure of a [topological space](https://bliptext.com/articles/topological-space), then it is customary to assume that is a [closed](https://bliptext.com/articles/closed-mathematics) subset of the topological [product space](https://bliptext.com/articles/product-space) P \\times P. Under this assumption partial order relations are well behaved at [limits](https://bliptext.com/articles/limit-of-a-sequence) in the sense that if and and for all i, then a \\leq b.\n\n# Intervals\nA **convex set** in a poset P is a subset I of P with the property that, for any x and y in I and any z in P, if x ≤ z ≤ y, then z is also in I. This definition generalizes the definition of [intervals](https://bliptext.com/articles/interval-mathematics) of [real numbers](https://bliptext.com/articles/real-number).\nWhen there is possible confusion with [convex sets](https://bliptext.com/articles/convex-set) of [geometry](https://bliptext.com/articles/geometry), one uses **order-convex** instead of \"convex\".\nA **convex sub[lattice](https://bliptext.com/articles/lattice-order-theory)** of a [lattice](https://bliptext.com/articles/lattice-order-theory) L is a sub[lattice](https://bliptext.com/articles/lattice-order-theory) of L that is also a convex set of L.\nEvery nonempty convex sublattice can be uniquely represented as the intersection of a [filter](https://bliptext.com/articles/filter-mathematics) and an [ideal](https://bliptext.com/articles/ideal-order-theory) of L.\nAn **interval** in a poset P is a subset that can be defined with **interval** notation:\n0 < x < 1\n.\nWhenever a ≤ b does not hold, all these intervals are empty.\nEvery interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set is convex, but not an interval.\nAn interval I is bounded if there exist elements a, b \\in P such that\nI ⊆\n. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true.\nFor example, let\nP = ∪ ∪\nas a subposet of the real numbers.\nThe subset is a bounded interval, but it has no [infimum](https://bliptext.com/articles/infimum) or [supremum](https://bliptext.com/articles/supremum) in P, so it cannot be written in interval notation using elements of P.\nA poset is called [locally finite](https://bliptext.com/articles/locally-finite-poset) if every bounded interval is finite.\nFor example, the integers are locally finite under their natural ordering.\nThe lexicographical order on the cartesian product is not locally finite, since\n(1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1)\n. Using the interval notation, the property \"a is covered by b\" can be rephrased equivalently as\nThis concept of an interval in a partial order should not be confused with the particular class of partial orders known as the [interval orders](https://bliptext.com/articles/interval-order).\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:45:23.233Z","updated_at":"2025-01-25T16:45:23.233Z","search_vector":"'order':2 'partial':1 'set':3","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), especially [order theory](https://bliptext.com/articles/order-theory), a **partial order** on a [set](https://bliptext.com/articles/set-mathematics) is an arrangement such that, for certain pairs of elements, one precedes the other.\nThe word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other.\nPartial orders thus generalize [total orders](https://bliptext.com/articles/total-order), in which every pair is comparable.\nFormally, a partial order is a [homogeneous binary relation](https://bliptext.com/articles/homogeneous-binary-relation) that is [reflexive](https://bliptext.com/articles/reflexive-relation), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), and [transitive](https://bliptext.com/articles/transitive-relation).\nA **partially ordered set** (**poset** for short) is an ordered pair P=(X,\\leq) consisting of a set X (called the ground set of P) and a partial order \\leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a **poset**.\n# [Fig. 1 The Hasse diagram of the set of all subsets of a three-element set ordered by inclusion. Sets connected by an upward path, like \\emptyset and \\{x,y\\}, are comparable, while e.g. \\{x\\} and \\{y\\} are not. | upload.wikimedia.org/wikipedia/commons/e/ea/Hasse///diagram///of///powerset///of///3.svg]\n\n# Partial order relations\nThe term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders.\nHowever some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders.\nStrict and non-strict partial orders can be put into a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence), so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.\n\n## Partial orders\n****A**** **************r****e****f****l****e****x****i****v****e**************,**** **************w****e****a****k**************,**** ****o****r**** ****,**** ****c****o****m****m****o****n****l****y**** ****r****e****f****e****r****r****e****d**** ****t****o**** ****s****i****m****p****l****y**** ****a****s**** ****a**** ****p****a****r****t****i****a****l**** ****o****r****d****e****r****,**** ****i****s**** ****a**** ****h****o****m****o****g****e****n****e****o****u****s**** ****r****e****l****a****t****i****o****n**** ****≤**** ****o****n**** ****a**** ****s****e****t**** ****P**** ****t****h****a****t**** ****i****s**** **************r****e****f****l****e****x****i****v****e**************,**** ****a****n****t****i****s****y****m****m****e****t****r****i****c****,**** ****a****n****d**** ****t****r****a****n****s****i****t****i****v****e****.****\nThat is, for all it must satisfy:\nA non-strict partial order is also known as an antisymmetric [preorder](https://bliptext.com/articles/preorder).\n\n## Strict partial orders\n****A****n**** **************i****r****r****e****f****l****e****x****i****v****e**************,**** **************s****t****r****o****n****g**************,**** ****o****r**** **** ****i****s**** ****a**** ****h****o****m****o****g****e****n****e****o****u****s**** ****r****e****l****a****t****i****o****n**** **** , where \\leq is a non-strict partial order relation on P, is the dual of .\nThe relation > is the converse of the irreflexive kernel of \\leq, which is always a subset of the complement of \\leq, but > is equal to the complement of \\leq [if, and only if](https://bliptext.com/articles/if-and-only-if), \\leq is a total order.\n\n# Alternative definitions\nAnother way of defining a partial order, found in [computer science](https://bliptext.com/articles/computer-science), is via a notion of [comparison](https://bliptext.com/articles/comparability).\nSpecifically, given as defined previously, it can be observed that two elements x and y may stand in any of four [mutually exclusive](https://bliptext.com/articles/mutually-exclusive) relationships to each other: either x y, or x and y are incomparable.\nThis can be represented by a function that returns one of four codes when given two elements.\nThis definition is equivalent to a partial order on a [setoid](https://bliptext.com/articles/setoid), where equality is taken to be a defined [equivalence relation](https://bliptext.com/articles/equivalence-relation) rather than set equality.\nWallis defines a more general notion of a partial order relation as any [homogeneous relation](https://bliptext.com/articles/homogeneous-relation) that is [transitive](https://bliptext.com/articles/transitive-relation) and [antisymmetric](https://bliptext.com/articles/antisymmetric-relation).\nThis includes both reflexive and irreflexive partial orders as subtypes.\nA finite poset can be visualized through its [Hasse diagram](https://bliptext.com/articles/hasse-diagram).\nSpecifically, taking a strict partial order relation (P,<), a [directed acyclic graph](https://bliptext.com/articles/directed-acyclic-graph) (DAG) may be constructed by taking each element of P to be a node and each element of < to be an edge.\nThe [transitive reduction](https://bliptext.com/articles/transitive-reduction) of this DAG is then the Hasse diagram.\nSimilarly this process can be reversed to construct strict partial orders from certain DAGs.\nIn contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.\n\n# Examples\nStandard examples of posets arising in mathematics include:\nOne familiar example of a partially ordered set is a collection of people ordered by [genealogical](https://bliptext.com/articles/genealogy) descendancy.\nSome pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.\n# [Fig. 3 Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4 | upload.wikimedia.org/wikipedia/commons/f/f3/Division///relation///4.svg]\n\n## Orders on the Cartesian product of partially ordered sets\nIn order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the [Cartesian product](https://bliptext.com/articles/cartesian-product) of two partially ordered sets are (see Fig. 4):\nAll three can similarly be defined for the Cartesian product of more than two sets.\nApplied to [ordered vector spaces](https://bliptext.com/articles/ordered-vector-space) over the same [field](https://bliptext.com/articles/field-mathematics), the result is in each case also an ordered vector space.\nSee also [orders on the Cartesian product of totally ordered sets](https://bliptext.com/articles/total-order).\n\n## Sums of partially ordered sets\nAnother way to combine two (disjoint) posets is the **ordinal sum** (or **linear sum**), Z = X ⊕ Y, defined on the union of the underlying sets X and Y by the order a ≤Z b if and only if:\nIf two posets are [well-ordered](https://bliptext.com/articles/well-ordered), then so is their ordinal sum.\n[Series-parallel partial orders](https://bliptext.com/articles/series-parallel-partial-order) are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition.\nParallel composition is the [disjoint union](https://bliptext.com/articles/disjoint-union) of two partially ordered sets, with no order relation between elements of one set and elements of the other set.\n\n# Derived notions\nThe examples use the poset consisting of the [set of all subsets](https://bliptext.com/articles/power-set) of a three-element set ordered by set inclusion (see Fig. 1).\n\n## Extrema\nThere are several notions of \"greatest\" and \"least\" element in a poset P, notably:\nAs another example, consider the positive [integers](https://bliptext.com/articles/integer), ordered by divisibility: 1 is a least element, as it [divides](https://bliptext.com/articles/divisor) all other elements; on the other hand this poset does not have a greatest element.\nThis partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal.\nIf 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 [prime number](https://bliptext.com/articles/prime-number) is a minimal element for it.\nIn this poset, 60 is an upper bound (though not a least upper bound) of the subset 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.\nIf the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig. 6).\n# [Fig. 5 The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all elements, and the bottom row are all elements, but there is no and no element. | upload.wikimedia.org/wikipedia/commons/9/9e/Hasse///diagram///of///powerset///of///3///no///greatest///or///least.svg]\n# [Fig. 6 Nonnegative integers, ordered by divisibility | upload.wikimedia.org/wikipedia/commons/e/e6/Infinite///lattice///of///divisors.svg]\n\n# Mappings between partially ordered sets\nGiven two partially ordered sets\n(S, ≤)\nand\n(T, ≼)\n, a function f : S \\to T is called **[order-preserving](https://bliptext.com/articles/order-preserving)**, or **[monotone](https://bliptext.com/articles/monotonic-function)**, or **isotone**, if for all x, y \\in S, x \\leq y implies\nf(x) ≼ f(y)\n. If\n(U, ≲)\nis also a partially ordered set, and both f : S \\to T and g : T \\to U are order-preserving, their [composition](https://bliptext.com/articles/function-composition) is order-preserving, too.\nA function f : S \\to T is called **order-reflecting** if for all x, y \\in S,\nf(x) ≼ f(y)\nimplies x \\leq y. If f is both order-preserving and order-reflecting, then it is called an **[order-embedding](https://bliptext.com/articles/order-embedding)** of\n(S, ≤)\ninto\n(T, ≼)\n. In the latter case, f is necessarily [injective](https://bliptext.com/articles/injective), since f(x) = f(y) implies and in turn x = y according to the antisymmetry of \\leq.\nIf an order-embedding between two posets S and T exists, one says that S can be **embedded** into T.\nIf an order-embedding f : S \\to T is [bijective](https://bliptext.com/articles/bijective), it is called an **[order isomorphism](https://bliptext.com/articles/order-isomorphism)**, and the partial orders\n(S, ≤)\nand\n(T, ≼)\nare said to be **isomorphic**.\nIsomorphic orders have structurally similar [Hasse diagrams](https://bliptext.com/articles/hasse-diagram) (see Fig. 7a).\nIt can be shown that if order-preserving maps f : S \\to T and g : T \\to U exist such that g \\circ f and f \\circ g yields the [identity function](https://bliptext.com/articles/identity-function) on S and T, respectively, then S and T are order-isomorphic.\nFor example, a mapping from the set of natural numbers (ordered by divisibility) to the [power set](https://bliptext.com/articles/power-set) of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its [prime divisors](https://bliptext.com/articles/prime-divisor).\nIt 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 12 does not divide 6).\nTaking instead each number to the set of its [prime power](https://bliptext.com/articles/prime-power) divisors defines a map that is order-preserving, order-reflecting, and hence an order-embedding.\nIt is not an order-isomorphism (since it, for instance, does not map any number to the set \\{4\\}), but it can be made one by [restricting its codomain](https://bliptext.com/articles/injective-function) to g(\\N).\nFig. 7b 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](https://bliptext.com/articles/distributive-lattice); see [Birkhoff's representation theorem](https://bliptext.com/articles/birkhoff-s-representation-theorem).\n\n# Number of partial orders\nSequence [ A001035] in [OEIS](https://bliptext.com/articles/on-line-encyclopedia-of-integer-sequences) gives the number of partial orders on a set of n labeled elements:\nThe number of strict partial orders is the same as that of partial orders.\nIf the count is made only [up to](https://bliptext.com/articles/up-to) isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.\n\n# Subposets\nA poset is called a **subposet** of another poset P=(X, \\leq) provided that X^* is a [subset](https://bliptext.com/articles/subset) of X and \\leq^* is a [subset](https://bliptext.com/articles/subset) of \\leq.\nThe latter condition is equivalent to the requirement that for any x and y in X^* (and thus also in X), if x \\leq^* y then x \\leq y.\nIf P^* is a subposet of P and furthermore, for all x and y in X^*, whenever x \\leq y we also have x \\leq^* y, then we call P^* the subposet of P **induced** by X^*, and write.\n\n# Linear extension\nA partial order \\leq^* on a set X is called an **extension** of another partial order \\leq on X provided that for all elements x, y \\in X, whenever x \\leq y, it is also the case that x \\leq^* y. A linear **extension** is an **extension** that is also a linear (that is, total) order.\nAs a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.\nEvery partial order can be extended to a total order ([order-extension principle](https://bliptext.com/articles/order-extension-principle)).\nIn [computer science](https://bliptext.com/articles/computer-science), algorithms for finding linear extensions of partial orders (represented as the [reachability](https://bliptext.com/articles/reachability) orders of [directed acyclic graphs](https://bliptext.com/articles/directed-acyclic-graph)) are called [topological sorting](https://bliptext.com/articles/topological-sorting).\n\n# In category theory\nEvery poset (and every [preordered set](https://bliptext.com/articles/preorder)) may be considered as a [category](https://bliptext.com/articles/category-mathematics) where, for objects x and y, there is at most one [morphism](https://bliptext.com/articles/morphism) from x to y. More explicitly, let hom(x, y) = if x ≤ y (and otherwise the [empty set](https://bliptext.com/articles/empty-set)) and Such categories are sometimes called [posetal](https://bliptext.com/articles/posetal-category).\nPosets are [equivalent](https://bliptext.com/articles/equivalence-of-categories) to one another if and only if they are [isomorphic](https://bliptext.com/articles/isomorphism-of-categories).\nIn a poset, the smallest element, if it exists, is an [initial object](https://bliptext.com/articles/initial-object), and the largest element, if it exists, is a [terminal object](https://bliptext.com/articles/terminal-object).\nAlso, every preordered set is equivalent to a poset.\nFinally, every subcategory of a poset is [isomorphism-closed](https://bliptext.com/articles/isomorphism-closed).\n\n# Partial orders in topological spaces\nIf P is a partially ordered set that has also been given the structure of a [topological space](https://bliptext.com/articles/topological-space), then it is customary to assume that is a [closed](https://bliptext.com/articles/closed-mathematics) subset of the topological [product space](https://bliptext.com/articles/product-space) P \\times P. Under this assumption partial order relations are well behaved at [limits](https://bliptext.com/articles/limit-of-a-sequence) in the sense that if and and for all i, then a \\leq b.\n\n# Intervals\nA **convex set** in a poset P is a subset I of P with the property that, for any x and y in I and any z in P, if x ≤ z ≤ y, then z is also in I. This definition generalizes the definition of [intervals](https://bliptext.com/articles/interval-mathematics) of [real numbers](https://bliptext.com/articles/real-number).\nWhen there is possible confusion with [convex sets](https://bliptext.com/articles/convex-set) of [geometry](https://bliptext.com/articles/geometry), one uses **order-convex** instead of \"convex\".\nA **convex sub[lattice](https://bliptext.com/articles/lattice-order-theory)** of a [lattice](https://bliptext.com/articles/lattice-order-theory) L is a sub[lattice](https://bliptext.com/articles/lattice-order-theory) of L that is also a convex set of L.\nEvery nonempty convex sublattice can be uniquely represented as the intersection of a [filter](https://bliptext.com/articles/filter-mathematics) and an [ideal](https://bliptext.com/articles/ideal-order-theory) of L.\nAn **interval** in a poset P is a subset that can be defined with **interval** notation:\n0 < x < 1\n.\nWhenever a ≤ b does not hold, all these intervals are empty.\nEvery interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set is convex, but not an interval.\nAn interval I is bounded if there exist elements a, b \\in P such that\nI ⊆\n. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true.\nFor example, let\nP = ∪ ∪\nas a subposet of the real numbers.\nThe subset is a bounded interval, but it has no [infimum](https://bliptext.com/articles/infimum) or [supremum](https://bliptext.com/articles/supremum) in P, so it cannot be written in interval notation using elements of P.\nA poset is called [locally finite](https://bliptext.com/articles/locally-finite-poset) if every bounded interval is finite.\nFor example, the integers are locally finite under their natural ordering.\nThe lexicographical order on the cartesian product is not locally finite, since\n(1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1)\n. Using the interval notation, the property \"a is covered by b\" can be rephrased equivalently as\nThis concept of an interval in a partial order should not be confused with the particular class of partial orders known as the [interval orders](https://bliptext.com/articles/interval-order).\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

History