Partially ordered set

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In mathematics, especially order theory, 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 binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

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 and , are comparable, while e.g. and are not.

Informal definitionEdit

A partial order defines a notion of comparison. Two elements x and y 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.

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, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.

A poset can be visualized through its Hasse diagram, which depicts the ordering relation.[1]

Partial order relationEdit

A partial order relation is a homogeneous relation that is transitive and antisymmetric.[2] There are two common sub-definitions for a partial order relation, for reflexive and irreflexive partial order relations, also called "non-strict" and "strict" respectively. The two definitions can be put into a one-to-one correspondence, so for every strict partial order there is a unique corresponding non-strict partial order, and vice-versa. The term partial order typically refers to a non-strict partial order relation.

Non-strict partial orderEdit

A reflexive, weak,[2] or non-strict partial order[3] is a homogeneous relation ≤ over a set   that is reflexive, antisymmetric, and transitive. That is, for all   it must satisfy:

  1. reflexivity:  , i.e. every element is related to itself.
  2. antisymmetry: if  , i.e. no two distinct elements precede each other.
  3. transitivity: if  .

A non-strict partial order is also known as an antisymmetric preorder.

Strict partial orderEdit

An irreflexive, strong,[2] or strict partial order on   is a homogeneous relation < on   that is irreflexive, transitive and asymmetric; that is, it satisfies the following conditions for all  

  1. Irreflexivity: not  , i.e. no element is related to itself
  2. Transitivity: if  
  3. Asymmetry: if   then not  .

Irreflexivity and transitivity together imply asymmetry. Also, asymmetry implies irreflexivity. In other words, a transitive relation is asymmetric if and only if it is irreflexive.[4] So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

A strict partial order is also known as a strict preorder.

Correspondence of strict and non-strict partial order relationsEdit

 
Fig.2 Commutative diagram about the connection between reflexive closure (cls), irreflexive kernel (ker), and converse relation (cnv), along an example relation (Hasse diagram depicted).

Strict and non-strict partial orders on a set   are closely related. A non-strict partial order   may be converted to a strict partial order by removing all relationships of the form   that is, the strict partial order is the set   where   is the identity relation on   and   denotes set subtraction. Conversely, a strict partial order < on   may be converted to a non-strict partial order by adjoining all relationships of that form; that is,   is a non-strict partial order. Thus, if   is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by

 
Conversely, if < is a strict partial order, then the corresponding non-strict partial order   is the reflexive closure given by:
 

Dual ordersEdit

The dual (or opposite)   of a partial order relation   is defined by letting   be the converse relation of  , i.e.   if and only if  . The dual of a non-strict partial order is a non-strict partial order,[5] and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.

NotationEdit

We can consider a poset as a 3-tuple  ,[6] or even a 5-tuple  ,[citation needed] where   and   are non-strict partial order relations,   and   are strict partial order relations, the dual of   is  , and   and   are likewise duals of each other.

Any one of the four partial order relations   on a given set uniquely determines the other three. Hence, as a matter of notation, we may write   or  , and assume that the other relations are defined appropriately. Defining via a non-strict partial order   is most common. Some authors use different symbols than   such as  [7] or  [8] to distinguish partial orders from total orders.

When referring to partial orders,   should not be taken as the complement of  .   is the converse of the irreflexive kernel of  , which is always a subset of the complement of  , but   is equal to the complement of   if, and only if,   is a total order.[a]

ExamplesEdit

Standard examples of posets arising in mathematics include:

  • The real numbers, or in general any totally ordered set, ordered by the standard less-than-or-equal relation ≤, is a non-strict partial order.
  • On the real numbers   the usual less than relation < is a strict partial order and the same is also true of the usual greater than relation > on  
  • By definition, every strict weak order is a strict partial order.
  • The set of subsets of a given set (its power set) ordered by inclusion (see Fig.1). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.
  • The set of natural numbers equipped with the relation of divisibility.
  • The vertex set of a directed acyclic graph ordered by reachability.
  • 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 sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally,   if and only if   for all  ; that is, a componentwise order.
  • For a set X and a partially ordered set P, the function space containing all functions from X to P, where fg if and only if f(x) ≤ g(x) for all  
  • 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 and, in most cases,[b] general relativity, where for two events X and Y, XY if and only if Y is in the future light cone of X. An event Y can only be causally affected by X if XY.

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

Orders on the Cartesian product of partially ordered setsEdit

Fig.3 Lexicographic order on  
Fig.4 Product order on  
Fig.5 Reflexive closure of strict direct product order on   Elements covered 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 Fig.3-5):

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.

See also orders on the Cartesian product of totally ordered sets.

Sums of partially ordered setsEdit

Another way to combine two (disjoint) posets is the ordinal sum[9] (or linear sum),[10] Z = XY, defined on the union of the underlying sets X and Y by the order aZ b if and only if:

  • a, bX with aX b, or
  • a, bY with aY b, or
  • aX and bY.

If two posets are well-ordered, then so is their ordinal sum.[11]

Series-parallel partial orders are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the disjoint union of two partially ordered sets, with no order relation between elements of one set and elements of the other set.

Derived notionsEdit

The examples use the poset   consisting of the set of all subsets of a three-element set   ordered by set inclusion (see Fig.1).

  • a is related to b when ab. This does not imply that b is also related to a, because the relation need not be symmetric. For example,   is related to   but not the reverse.
  • a and b are comparable if ab or ba. Otherwise they are incomparable. For example,   and   are comparable, while   and   are not.
  • A total order or linear order is a partial order under which every pair of elements is comparable, i.e. trichotomy holds. For example, the natural numbers with their standard order.
  • A chain is a subset of a poset that is a totally ordered set. For example,   is a chain.
  • An antichain is a subset of a poset in which no two distinct elements are comparable. For example, the set of singletons  
  • An element a is said to be strictly less than an element b, if ab and   For example,   is strictly less than  
  • An element a is said to be covered by another element b, written ab (or a <: b), if a is strictly less than b and no third element c fits between them; formally: if both ab and   are true, and acb is false for each c with   Using the strict order <, the relation ab can be equivalently rephrased as "a < b but not a < c < b for any c". For example,   is covered by  but is not covered by  

ExtremaEdit

 
Fig.6 The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all maximal elements, and the bottom row are all minimal elements, but there is no greatest and no least element.

There are several notions of "greatest" and "least" element in a poset   notably:

  • Greatest element and least element: An element   is a greatest element if for every element   An element   is a least element if for every element   A poset can only have one greatest or least element. In our running example, the set   is the greatest element, and   is the least.
  • Maximal elements and minimal elements: An element   is a maximal element if there is no element   such that   Similarly, an element   is a minimal element if there is no element   such that   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. In our running example,   and   are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig.6).
  • 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. In our example, the set   is an upper bound for the collection of elements  
 
Fig.7 Nonnegative integers, ordered by divisibility

As another example, consider the positive integers, 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 Fig.7). This 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. 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 prime number is a minimal element for it. In 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.

Mappings between partially ordered setsEdit

Fig.8 Order-preserving, but not order-reflecting (since f(u) ≼ f(v), but not u   v) map.
Fig.9 Order isomorphism between the divisors of 120 (partially ordered by divisibility) and the divisor-closed subsets of {2, 3, 4, 5, 8} (partially ordered by set inclusion)

Given two partially ordered sets (S, ≤) and (T, ≼),[c] a function   is called order-preserving, or monotone, or isotone, if for all     implies f(x) ≼ f(y). If (U, ≲) is also a partially ordered set, and both   and   are order-preserving, their composition   is order-preserving, too. A function   is called order-reflecting if for all   f(x) ≼ f(y) implies   If   is both order-preserving and order-reflecting, then it is called an order-embedding of (S, ≤) into (T, ≼). In the latter case,   is necessarily injective, since   implies   and in turn   according to the antisymmetry of   If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding   is bijective, it is called an order isomorphism, and the partial orders (S, ≤) and (T, ≼) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams (see Fig.8). It can be shown that if order-preserving maps   and   exist such that   and   yields the identity function on S and T, respectively, then S and T are order-isomorphic.[12]

For example, a mapping   from the set of natural numbers (ordered by divisibility) to the power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. It is order-preserving: if   divides   then each prime divisor of   is also a prime divisor of   However, it is neither injective (since it maps both 12 and 6 to  ) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its prime power divisors defines a map   that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set  ), but it can be made one by restricting its codomain to   Fig.9 shows a subset of   and its isomorphic image under   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 ordersEdit

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:

Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Preorder Partial order Total preorder Total order Equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65,536 3,994 4,096 355 219 75 24 15
n 2n2 2n2n  S(n, k) n!  S(n, k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

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

If the count is made only up to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... (sequence A000112 in the OEIS) is obtained.

Linear extensionEdit

A partial order   on a set   is an extension of another partial order   on   provided that for all elements   whenever   it is also the case that   A linear extension is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order (order-extension principle).[13]

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

Directed acyclic graphsEdit

Strict partial orders correspond directly to directed acyclic graphs (DAGs). If a graph is constructed by taking each element of   to be a node and each element of   to be an edge, then 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. In contrast a non-strict partial order would have self loops at every node and therefore not be a DAG.

In category theoryEdit

Every poset (and every preordered set) may be considered as a category where, for objects   and   there is at most one morphism from   to   More explicitly, let hom(x, y) = {(x, y)} if xy (and otherwise the empty set) and   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 initial object, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

Partial orders in topological spacesEdit

If   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 product space   Under this assumption partial order relations are well behaved at limits in the sense that if   and   and for all     then  [14]

IntervalsEdit

An interval 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 xzy, then z is also in I. (This definition generalizes the interval definition for real numbers.)

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

Using the corresponding strict relation "<", the open interval (a, b) 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 (1, 2) on the integers is empty since there are no integers I such that 1 < I < 2.

The half-open intervals [a, b) and (a, b] are defined similarly.

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

An interval I is bounded if there exist elements   such that I[a, b]. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let P = (0, 1)(1, 2)(2, 3) as a subposet of the real numbers. The subset (1, 2) is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P.

A poset is called locally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product   is not locally finite, since (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  

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

See alsoEdit

NotesEdit

  1. ^ A proof can be found here.
  2. ^ See General relativity#Time travel.
  3. ^ The partial orders   and ≼ can be different, but need not.

CitationsEdit

  1. ^ Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. pp. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hasse diagram...
  2. ^ a b c Wallis, W. D. (14 March 2013). A Beginner’s Guide to Discrete Mathematics. Springer Science & Business Media. p. 100. ISBN 978-1-4757-3826-1.
  3. ^ Simovici, Dan A. & Djeraba, Chabane (2008). "Partially Ordered Sets". Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer. ISBN 9781848002012.
  4. ^ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I. Prague: School of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  5. ^ Davey & Priestley (2002), pp. 14-15.
  6. ^ Avigad, Jeremy; Lewis, Robert Y.; van Doorn, Floris (29 March 2021). "13.2. More on Orderings". Logic and Proof (Release 3.18.4 ed.). Retrieved 24 July 2021. So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one.
  7. ^ Rounds, William C. (7 March 2002). "Lectures slides" (PDF). EECS 203: DISCRETE MATHEMATICS. Retrieved 23 July 2021.
  8. ^ Kwong, Harris (25 April 2018). "7.4: Partial and Total Ordering". A Spiral Workbook for Discrete Mathematics. Retrieved 23 July 2021.
  9. ^ Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 62–63, ISBN 9789810235895
  10. ^ Davey & Priestley (2002), pp. 17–18.
  11. ^ P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.
  12. ^ Davey & Priestley (2002), pp. 23–24.
  13. ^ Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.
  14. ^ Ward, L. E. Jr (1954). "Partially Ordered Topological Spaces". Proceedings of the American Mathematical Society. 5 (1): 144–161. doi:10.1090/S0002-9939-1954-0063016-5. hdl:10338.dmlcz/101379.

ReferencesEdit

External linksEdit