Posets

Partially ordered sets for Julia based on Graphs. See HasseDiagrams for functions to draw posets.

Introduction: Partially Ordered Sets

A partially ordered set, or poset for short, is a pair $(V,\prec)$ where $V$ is a set and $\prec$ is a binary relation on $V$ that is

irreflexive (for all $v \in V$, it is never the case that $v \prec v$),
antisymmetric (for all $v,w \in V$, we never have both $v \prec w$ and $w \prec v$), and
transitive (for all $u,v,w \in V$, if $u \prec v$ and $v \prec w$ then $u \prec w$).

Posets are naturally represented as transitively closed, directed, acyclic graphs. This is how this module implements posets using the DiGraph type in Graphs.

The design philosophy for this module is modeled exactly on Graphs. In particular, the vertex set of a poset is necessarily of the form {1,2,...,n}.

Basics

Constructing new posets

Create a new poset with no elements using Poset() or a poset with a specified number of elements with Poset(n).

Given a poset p, use Poset(p) to create an independent copy of p.

Given a directed graph d, use Poset(d) to create a new poset from the transitive closure of d. An error is thrown if d has cycles. (Self loops in d are ignored.)

Given a square matrix A, create a poset in which i < j exactly when the i,j-entry of A is nonzero. Diagonal entries are ignored. If this matrix would create a cycle, an error is thrown.

Adding vertices (elements)

For consistency with Graph, we call the elements of a Poset vertices and the functions add_vertex! and add_vertices! work exactly as in the Graphs module.

julia> using Posets

julia> p = Poset()
{0, 0} Int64 poset

julia> add_vertex!(p)
true

julia> add_vertices!(p,5)
5

julia> p
{6, 0} Int64 poset

Use nv(p) to return the number of elements (vertices) in p.

Adding a relation

To add a relation to a poset, use add_relation!. This returns true when successful.

julia> p = Poset(4)
{4, 0} Int64 poset

julia> add_relation!(p,1,2)
true

julia> add_relation!(p,2,3)
true

julia> add_relation!(p,3,1)
false

Let's look at this carefully to understand why the third call to add_relation! does not succeed:

The first call to add_relation! causes the relation 1 < 2 to hold in p.
The second call to add_relation! causes the relation 2 < 3 to be added to p. Given that 1 < 2 and 2 < 3, by transitivity we automatically have 1 < 3 in p.
Therefore, we cannot add 3 < 1 as a relation to this poset as that would violate antisymmetry.

The add_relation! function may also be called as add_relation!(p, (a,b)) or add_relation!(p, a => b). Both are equivalent to add_relations(p, a, b).

Adding multiple relations (Danger!)

The addition of a relation to a poset can be somewhat slow. Each addition involves error checking and calculations to ensure the integrity of the underlying data structure. See the Implementation section at the end of this document. Adding a list of relations one at a time can be inefficient, but it is safe. We also provide the function add_relations! (plural) that is more efficient, but can cause serious problems.

To underscore the risk, this function is not exported, but needs to be invoked as Posets.add_relations!(p, rlist) where rlist is a list of either tuples (a,b) or pairs a => b.

Here is a good application of this function (although using chain(10) is safer):

julia> p = Poset(10)
{10, 0} Int64 poset

julia> rlist = ((i,i+1) for i=1:9)
Base.Generator{UnitRange{Int64}, var"#13#14"}(var"#13#14"(), 1:9)

julia> Posets.add_relations!(p, rlist)

julia> p == chain(10)
true

Here is what happens with misuse:

julia> p = Poset(5)
{5, 0} Int64 poset

julia> rlist = [ 1=>2, 2=>3, 3=>1 ]
3-element Vector{Pair{Int64, Int64}}:
 1 => 2
 2 => 3
 3 => 1

julia> Posets.add_relations!(p, rlist)
ERROR: This poset has been become corrupted!

Removing an element

The function rem_vertex! behaves exactly as in Graphs. It removes the given vertex from the poset. For example:

julia> p = Poset(5)
{5, 0} Int64 poset

julia> add_relation!(p,1,5)
true

julia> rem_vertex!(p,2)
true

julia> has_relation(p,1,2)
true

When element 2 is removed from p, element 5 takes its place. Because of this renumbering, we have some unexpected behavior:

julia> p = subset_lattice(4)
{16, 65} Int64 poset

julia> q = Poset(p)   # make a copy of p
{16, 65} Int64 poset

julia> rem_vertex!(q, 9)
true

julia> q
{15, 57} Int64 poset

julia> q ⊆ p
false

julia> maximals(p) |> collect
1-element Vector{Int64}:
 16

julia> maximals(q) |> collect
1-element Vector{Int64}:
 9

One might expect that deleting a vertex from a poset results in a poset that is a subset of the original. However, when vertex 9 was removed from (a copy of) p, the vertex 16 is relabeled 9. Hence vertex 9 in p is not maximal, but it is maximal in q.

For a more extensive explanation, see poset-deletion.pdf in the delete-doc folder.

Removing a relation

Removing relations from a poset is accomplished with rem_relation!(p,a,b). Assuming a<b in p, this deletes the relation a<b from p, but also deletes all relations a<x and x<b for vertices x that lie between a and b.

julia> p = chain(5)
{5, 10} Int64 poset

julia> rem_relation!(p, 2, 4)
true

julia> collect(relations(p))
8-element Vector{Relation{Int64}}:
 Relation 1 < 2
 Relation 1 < 3
 Relation 1 < 4
 Relation 1 < 5
 Relation 2 < 4
 Relation 2 < 5
 Relation 3 < 5
 Relation 4 < 5

Note that relations 2<3 and 3<4 have been removed.

For a more extensive explanation, see poset-deletion.pdf in the delete-doc folder.

`poset_builder` function

Posets are a collection of objects and a $\prec$ relation. The poset_builder function takes a list of objects and a function the determines if they obey a $\prec$ relation, and returns the resulting poset.

The function is invoked as poset_builder(list, comp_func) where

list is a list (vector) of objects and
comp_func is a function that takes a pair of objects and returns true if they satisfy a $\prec$ relation.

For example, suppose we want to construct a poset consisting of the integers 1 to 100 which $a \prec b$ provided $a$ is a divisor of $b$. Here is how to do that:

julia> nums = collect(1:100);

julia> divs(a,b) = mod(b,a)==0    # return true if a divides b
divs (generic function with 1 method)

julia> p = poset_builder(nums, divs)
{100, 382} Int64 poset

The elements that cover 1 are the primes less than 100:

julia> [ x for x in 1:100 if covered_by(p,1,x) ]
25-element Vector{Int64}:
  2
  3
  5
  7
 11
 13
 17
 19
  ⋮
 67
 71
 73
 79
 83
 89
 97

Inspection

Vertices

Use nv(p) to return the number of vertices in the poset p. As in Graphs, the elements of the poset are integers from 1 to n.

Use in(a, p) [or a ∈ p] to determine if a is an element of p. This is equivalent to 1 <= a <= nv(p).

Relations

There are three ways to check if elements are related in a poset.

First, to see if 1 < 3 in p we use the has_relation function:

julia> has_relation(p,1,3)
true

Second, the syntax p(a,b) is equivalent to has_relation(p,a,b):

julia> p(1,3)
true

julia> p(3,1)
false

There is a third way to determine the relation between elements a and b in a poset p. Instead of has_relation(p,a,b) or p(a,b) we may use this instead: p[a] < p[b].

julia> has_relation(p,1,3)
true

julia> p[1] < p[3]
true

julia> p[3] < p[1]
false

The other comparison operators (<=, >, >=, ==, !=) works as expected.

julia> p[3] > p[1]
true

Neither has_relation(p,a,b) nor p(a,b) generate errors; they return false even if a or b are not elements of p.

julia> p(-2,9)
false

However, the expression p[a] < p[b] throws an error in either of these situations:

Using the syntax p[a] if a is not an element of p.
Trying to compare elements of different posets (even if the two posets are equal).

Comparability check

The functions are_comparable(p,a,b) and are_incomparable(p,a,b) behave as follows:

are_comparable(p,a,b) returns true exactly when a and b are both in the poset, and one of the following is true: a<b, a==b, or a>b.
are_incompable(p,a,b) returns true exactly when a and b are both in the poset, but none of the follower are true: a<b, a==b, or a>b.

Alternatively, use p[a] ⟂ p[b] to test if a and b are comparable, and use p[a] ∥ p[b] to test if a and b are incomparable.

Chain/antichain check

Given a list of elements vlist of a poset p:

is_chain(p, vlist) returns true if the elements of vlist form a chain in p.
is_antichain(p, vlist) returns true if the elements of vlist form an antichain in p.

Both return false if an element of vlist is not in p.

Counting/listing relations

Use nr to return the number of relations in the poset (this is analogous to ne in Graphs):

julia> nr(p)
3

The function relations returns an iterator for all the relations in a poset.

julia> p = chain(4)
{4, 6} Int64 poset

julia> collect(relations(p))
6-element Vector{Relation{Int64}}:
 Relation 1 < 2
 Relation 1 < 3
 Relation 1 < 4
 Relation 2 < 3
 Relation 2 < 4
 Relation 3 < 4

The functions src and dst return the lesser and greater elements of a relation, respectively:

julia> r = first(relations(p))
Relation 1 < 2

julia> src(r), dst(r)
(1, 2)

Subset

issubset(p,q) (or p ⊆ q) returns true exactly when nv(p) ≤ nv(q) and whenever v < w in p we also have v < w in q.

Above, below, between

above(p,a) returns an iterator for all elements k of p such that a<k.
below(p,a) returns an iterator for all elements k of p such that k<a.
between(p,a,b) returns an iterator for all elements k of p such that a<k<b.

julia> p = chain(10)
{10, 45} Int64 poset

julia> collect(above(p,6))
4-element Vector{Int64}:
  7
  8
  9
 10

julia> collect(below(p,6))
5-element Vector{Int64}:
 1
 2
 3
 4
 5

julia> collect(between(p,3,7))
3-element Vector{Int64}:
 4
 5
 6

Covers

In a poset, we say a is covered by b provided a < b and there is no element c such that a < c < b.

Use covered_by(p,a,b) to determine if a is covered by b. Alternatively, use p[a] << p[b] or p[b] >> p[a].

julia> p = chain(8)
{8, 28} Int64 poset

julia> p[4] << p[5]
true

julia> p[4] << p[6]
false

The functions just_above and just_below can be used to find elements that cover, or are covered by, a given vertex.

julia> p = chain(9)
{9, 36} Int64 poset

julia> above(p,5) |> collect
4-element Vector{Int64}:
 6
 7
 8
 9

julia> just_above(p,5) |> collect
1-element Vector{Int64}:
 6

julia> below(p,5) |> collect
4-element Vector{Int64}:
 1
 2
 3
 4

julia> just_below(p,5) |> collect
1-element Vector{Int64}:
 4

Maximals, minimals, height, and width

maximals(p) returns an iterator for the maximal elements of p.
minimals(p) returns an iterator for the minimal elements of p.
maximum(p) returns the maximum element of p or 0 if no such element exists.
minimum(p) returns the minimum element of p or 0 if no such element exists.
max_chain(p) returns a vector containing the elements of a largest chain in p.
max_antichain(p) returns a vector containing the elements of a largest antichain in p.
height(p) returns the size of a largest chain in p.
width(p) returns the size of a largest antichain in p.
chain_cover(p) returns a minimum-size collection of chains of p such that every element of p is in one of the chains. The number of chains is the width of p.
antichain_cover(p) returns a minimum-size collection of antichains of p such that every element of p is in one of the antichains. The number of antichains is the height of p.

Note: The function max_chain returns a largest chain in the poset. It is possible that there are two or more possible answers because there are two or more such chains of maximum size. There is no guarantee as to which largest chain will be returned. Likewise for max_antichain. Similarly, chain_cover returns a minimum-size partition of the elements into chains. If there are multiple minimum-size chain covers, there is no guarantee which will be returned by chain_cover. Likewise for antichain_cover.

Isomorphism

For posets p and q, use iso(p,q) to compute an isomorphism from p to q, or throw an error if the posets are not isomorphic.

Let f = iso(p,q). Then f is a Dict mapping vertices of p to vertices of q. For example, if p has a unique minimal element x, then f[x] is the unique minimal element of q.

To check if posets are isomorphic, use iso_check (which calls iso inside a try/catch block).

Realizers and dimension

A realizer for a poset p is a set of linear extensions whose intersection is p. The function realizer(p, d) returns a list of d linear extensions (total orders) that form a realizer of p, or throws an error if no realizer of that size exists.

julia> p = standard_example(3)
{6, 6} Int64 poset

julia> r = realizer(p, 3)
3-element Vector{Poset{Int64}}:
 {6, 15} Int64 poset
 {6, 15} Int64 poset
 {6, 15} Int64 poset

julia> r[1] ∩ r[2] ∩ r[3] == p
true

julia> realizer(p, 2)
ERROR: This poset has dimension greater than 2; no realizer found.

The dimension of a poset is the size of a smallest realizer. Use dimension(p) to calculate its dimension.

julia> p = standard_example(4)
{8, 12} Int64 poset

julia> dimension(p)
4

Note: Computation of the dimension of a poset is NP-hard. The dimension function may be slow, even for moderate-size posets.

Ranking

The function ranking(p) returns a ranking (as a dictionary) of the poset starting from the minimal elements. That is, the minimal elements are assigned rank 0. Elements that are above only elements of rank 0 are assigned rank 1. Elements that are only above elements of rank 0 and 1 are assigned rank 2. And so forth.

Note that not all posets are ranked, but this function always returns a result.

We also provide dual_ranking which combines the rankings of p and its dual, p'. This removes the bias that minimal elements are treated differently than maximal elements.

For example, suppose the poset is a the disjoint union of a 3-element and a 2-element chain. The function ranking places both minimal elements at rank 0, then their covers at rank 1, and finally the maximal element in the 3-chain at rank 2.

By contrast, dual_ranking places the elements in the 3-chain at ranks 0, 2, and 4, and the elements of the 2-chain at ranks 1 and 3.

julia> p = chain(3) + chain(2)
{5, 4} Int64 poset

julia> ranking(p)
Dict{Int64, Int64} with 5 entries:
  5 => 1
  4 => 0
  2 => 1
  3 => 2
  1 => 0

julia> dual_ranking(p)
Dict{Int64, Int64} with 5 entries:
  5 => 3
  4 => 1
  2 => 2
  3 => 4
  1 => 0

Standard Posets

The following functions create standard partially ordered sets. See the Gallery for pictures of some of these posets.

antichain(n) creates the poset with n elements and no relations. Same as Poset(n).
chain(n) creates the poset with n elements in which 1 < 2 < 3 < ... < n.
chain(vlist) creates a chain from the integer vector vlist (which must be a permutation of 1:n). For example, chain([2,1,3]) creates a chain in which 2 < 1 < 3.
chevron() creates a poset with 6 elements that has dimension equal to 3. It is different from standard_example(3).
crown(n,k) creates the crown poset with 2n elements with two levels: n elements as minimals and n as maximals. Each minimal is comparable to n-k maximals. See the help message for more information.
random_linear_order(n): Create a linear order in which the numbers 1 through n appear in random order.
random_poset(n,d=2): Create a random d-dimensional poset by intersecting d random linear orders, each with n elements.
standard_example(n) creates a poset with 2n elements. Elements 1 through n form an antichain as do elements n+1 through 2n. The only relations are of the form j < k where 1 ≤ j ≤ n and k = n+i where 1 ≤ i ≤ n and i ≠ j. This is a smallest-size poset of dimension n. Equivalent to crown(n,1).
subset_lattice(d): Create the poset corresponding to the 2^d subsets of {1,2,...,d} ordered by inclusion. For a between 1 and 2^d, element a corresponds to a subset of {1,2,...,d} as follows: Write a-1 in binary and view the bits as the characteristic vector indicating the members of the set. For example, if a equals 12, then a-1 is 1011 in binary. Reading off the digits from the right, this gives the set {1,2,4}.
- Use subset_decode(a) to convert an element a of this poset into a set of positive integers, A.
- Use subset_encode(A) to convert a set of positive integers to its name in this poset.
weak_order(vals): Create a weak order p from a list of real numbers. In p element i is less than element j provided vals[i] < vals[j] .

Graphs and Posets

Let p be a poset. The following two functions create graphs from p with the same vertex set as p:

comparability_graph(p) creates an undirected graph in which there is an edge from v to w exactly when v < w or w < v in p.
cover_digraph(p) creates a directed graph in which there is an edge from v to w exactly when v is covered by w.

julia> p = chain(9)
{9, 36} Int64 poset

julia> g = comparability_graph(p)
{9, 36} undirected simple Int64 graph

julia> g == complete_graph(9)
true

julia> d = cover_digraph(p)
{9, 8} directed simple Int64 graph

julia> d == path_digraph(9)
true

Given a graph g, calling incidence_poset(p) creates a poset whose elements correspond to the vertices and edges of g. In this poset the only relations are of the form v < e where v is a vertex that is an end point of the edge e.

Deprecation notice: This function was previously named vertex_edge_incidence_poset and that name is now deprecated. It will be removed in future releases.

Matrices

zeta_matrix(p) returns the zeta matrix of the poset. This is a 0,1-matrix whose i,j-entry is 1 exactly when p[i] ≤ p[j].
strict_zeta_matrix(p) returns a 0,1-matrix whose i,j entry is 1 exactly when p[i] < p[j].
mobius_matrix(p) returns the inverse of zeta(p).

In all cases, the output is a dense, integer matrix.

Operations

Dual

The dual of poset p is created using reverse(p). This returns a new poset with the same elements as p in which all relations are reversed (i.e., v < w in p if and only if w < v in reverse(p)). The dual (reverse) of p can also be created with p'.

Disjoint union

Given two posets p and q, the result of p+q is a new poset formed from the disjoint union of p and q. Note that p+q and q+p are isomorphic, but may be unequal because of the vertex numbering convention.

Alternatively hcat(p,q).

Stack

Given two posets p and q, the result of p/q is a new poset from a copy of p and a copy of q with all elements of p above all elements of q.

Alternatively, vcat(p,q) or q\p.

Cartesian product

Given posets $P$ and $Q$, their Cartesian product, $P \times Q$, is a poset whose elements are all ordered pairs $(a,b)$ where $a$ is an element of $P$ and $b$ is an element of $Q$. In this poset we have $(a,b)\preceq(c,d)$ if and only if $a\preceq c$ in $P$ and $b\preceq d$ in $Q$.

Cartesian product is implemented in Posets as p * q. The result is a new poset that is isomorphic to the Cartesian product of p and q.

Induced subposet

Given a poset p and a list of vertices vlist, use induced_subposet(p) to return a pair (q,vmap). The poset q is the induced subposet and the vector vmap maps the new vertices to the old ones (the vertex i in the subposet corresponds to the vertex vmap[i] in p).

This is exactly analogous to Graphs.induced_subgraph.

Intersection

Given two posets p and q, intersect(p,q) is a new poset in which v < w if and only if v < w in both p and q. The number of elements is the smaller of nv(p) and nv(q). This may also be invoked as p ∩ q.

For example, the intersection of a chain with its reversal has no relations:

julia> p = chain(5)
{5, 10} Int64 poset

julia> p ∩ reverse(p)
{5, 0} Int64 poset

Linear extension

Use linear_extension(p) to create a linear extension of p. This is a total order q with the same elements as p and with p ⊆ q.

The function random_linear_extension(p) creates a pseudo-random linear extension of p
as follows: Pick a minimal element of p at random. Make that the first element of the result. Now ignore that element and pick a minimal element at random from among those that remain. Continue in this way until all elements have been incorporated.

Join and meet

Let $x$ and $y$ be elements of a poset $P$. Let $U$ be the set of all elements $z$ of $P$ such that $x \preceq z$ and $y \preceq z$. This is the set of all elements above or equal to both $x$ and $y$. If $U$ contains a minimum element (one that is below all the other elements of $U$), then that minimum element $u$ is the join of $x$ and $y$. Notation $u = x \vee y$.

Similarly, let $D$ be the set of all elements $z$ of $P$ such that $z \preceq x$ and $z \preceq y$. This is the set of all elements in $P$ that are below or equal to $x$ and $y$. If $D$ contains a unique maximum element (one that is above all the other elements in $D$), then that maximum element $d$ is the meet of $x$ and $y$. Notation: $d = x \wedge y$.

There are two ways to compute the join [or meet] of elements in a poset.

The join and meet of elements x and y in poset p can be computed as p[x] ∨ p[y] and p[x] ∧ p[y].
Or use the functions lattice_join(p,x,y) or lattice_meet(p,x,y)

Important notes:

The meet [or join] of two elements need not exist.
- In the operation form, p[x] ∧ p[y] [or p[x] ∨ p[y]], if there is no meet [or join], an error is thrown.
- In the function form, lattice_meet(p,x,y) [or lattice_join(x,y)], if there is no meet [or join] then 0 is returned.
Cannot compute the meet [or join] of elements in different posets.
The expression p[x] throws an error if x is not an element of p.
The symbol ∨ is typed \vee<TAB> and ∧ is typed \wedge<TAB>.
The result of p[x] ∨ p[y] is an object of type PosetElement (likewise for ∧).
The result of lattice_join [or lattice_meet] is always an integer. The return value of 0 is used to show that the join [or meet] does not exist.

The join and meet operations for posets are analogous to union and intersection for sets as illustrated here:

julia> using ShowSet

julia> p = subset_lattice(4)
{16, 65} Int64 poset

julia> A = Set([1,2,3])
{1,2,3}

julia> B = Set([2,3,4])
{2,3,4}

julia> a = subset_encode(A)
8

julia> b = subset_encode(B)
15

julia> p[a] ∨ p[b]
Element 16 in a {16, 65} Int64 poset

julia> subset_decode(integer(ans))
{1,2,3,4}

julia> p[a] ∧ p[b]
Element 7 in a {16, 65} Int64 poset

julia> subset_decode(integer(ans))
{2,3}

Further, the function join_table(p) [or meet_table] creates a matrix whose i,j-entry is the join [or meet] of elements i and j in poset p, or 0 if the join [or meet] doesn't exist.

Implementation

A Poset is a structure that contains a single data element: a DiGraph. Users should not be accessing this directly, but it may be useful to understand how posets are implemented. The directed graph is acyclic (including loopless) and transitively closed. This means if $a \to b$ is an edge and $b\to c$ is an edge, then $a \to c$ is also an edge. The advantage to this structure is that checking if $a \prec b$ in a poset is quick. There are two disadvantages.

First, the graph may be larger than needed. If we only kept cover edges (the transitive reduction of the digraph) we might have many fewer edges. For example, a linear order with $n$ elements has $\binom{n}{2} \sim n^2/2$ edges in the digraph that represents it, whereas there are only $n-1$ edges in the cover digraph. However, this savings is an extreme example. A poset with $n$ elements split into two antichains, with every element of the first antichain below every element of the second, has $n^2/4$ edges in either representation. So in either case, the representing digraph may have up to order $n^2$ edges.

Second, the computational cost of adding (or deleting) a relation is nontrivial. The add_relation! function first checks if the added relation would violate transitivity; this is speedy because we can add the relation $a \prec b$ so long as we don't have $b\prec a$ already in the poset. However, after the edge $(a,b)$ is inserted into the digraph, we execute transitiveclosure! and that takes some work. Adding several relations to the poset, one at a time, can be slow.

This can be greatly accelerated by using Posets.add_relations! but (as discussed above) this function can cause severe problems if not used carefully.

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