FinitePosets

# FinitePosets — Module.

This package deals with finite posets.

There are two types of posets. A "canonical poset" or CPoset is on the elements 1:n where n=length(P). A Poset is on a given list of elements which can be of any type. A Poset internally contains aCPoset which works on the indices of the elements, which is more efficient than working with the elements themselves. For efficiency, many functions work on the internal CPoset by transforming their input to indices and their output to elements.

A CPoset p contains one of the following data:

hasse(p): a list representing the Hasse diagram of the poset: the i-th entry is the list of elements which cover (are immediate successors of) i, that is the list of j such that i<j and there is no k such that i<k<j.
incidence(p): a boolean matrix such that incidence[i,j]==true iff i<=j. This is sometimes called the ζ-matrix of the poset.

Some computations work better on the incidence matrix, and some others on the Hasse diagram. If needed for a computation and missing, one of the above data is computed from the other. This may take some substantial time for large posets.

There are several ways of defining a poset. One way is entering the Hasse diagram:

julia> p=CPoset([[2,3],[4],[4],Int[]])
1<2,3<4

As seen above, p is displayed as a list of covering maximal chains; elements which are equivalent for the poset are printed together separated by commas.

julia> length(p) # the number of elements of `p`
4

julia> incidence(p)
4×4 Matrix{Bool}:
 1  1  1  1
 0  1  0  1
 0  0  1  1
 0  0  0  1

julia> linear_extension(p) # a total order compatible with p
4-element Vector{Int64}:
 1
 2
 3
 4

A Poset is constructed from a CPoset and a list of elements

julia> P=Poset(p,[:a,:b,:c,:d])
a<b,c<d

julia> P.C # the CPoset attached to P
1<2,3<4

A convenient constructor for Posets takes a function representing isless for the poset and the list of elements and constructs the poset from the incidence matrix, computed by applying the function to each pair of elements. For isless one can give either a function implementing < or a function implementing ≤ (it is and-ed with != in any case).

julia> l=vec(collect(Iterators.product(1:2,1:2)))
4-element Vector{Tuple{Int64, Int64}}:
 (1, 1)
 (2, 1)
 (1, 2)
 (2, 2)

julia> P=Poset((x,y)->all(map(<=,x,y)),l)
(1, 1)<(2, 1),(1, 2)<(2, 2)

julia> eltype(P) # the type of the elements of P
Tuple{Int64, Int64}

julia> summary(P) # useful for big posets
"Poset{Tuple{Int64, Int64}} of length 4"

A poset can also be constructed from an incidence matrix so the last example could also be entered as

julia> P=Poset(CPoset([all(map(<=,x,y)) for x in l, y in l]),l)
(1, 1)<(2, 1),(1, 2)<(2, 2)

Flexibility on printing a Poset is obtained by setting the function show_element which takes as arguments an IO, the poset, and the index of the element to print:

julia> P.show_element=(io,p,n)->join(io,p.elements[n],".");

julia> P
1.1<2.1,1.2<2.2

julia> delete!(P,:show_element); # back to default

The above fancy printing applies only when printing at the REPL or in pluto or Jupyter. The default printing gives a form which can be input back in Julia

julia> print(P) 
Poset(CPoset([[2, 3], [4], [4], Int64[]]),[(1, 1), (2, 1), (1, 2), (2, 2)])

A poset can be specified by a list of tuples specifying order relations. The transitive closure of these relations is computed, resulting in an incidence matrix from which the poset is constructed. The elements of the poset, if not specified separately, are all the elements that appear in the tuples.

julia> Poset([(:a,:b),(:c,:d)])
a<b
c<d

julia> CPoset([(1,3),(2,5)]) # the CPoset is on 1:maximum(entries)
4
1<3
2<5

To get the order relation ≤ of the poset p between elements i and j just call ≤(p,i,j).

julia> ≤(P,(1,1),(2,1))
true

julia> ≤(P.C,1,2) # the same
true

Intervals in a poset can be computed with strict or not bounds.

julia> interval(P,≤,(1,2)) # elements below (1,2)
2-element Vector{Tuple{Int64, Int64}}:
 (1, 1)
 (1, 2)

julia> interval(P,≥,(1,2)) # elements above (1,2)
2-element Vector{Tuple{Int64, Int64}}:
 (1, 2)
 (2, 2)

julia> interval(P,<,(1,2)) # elements strictly below (1,2)
1-element Vector{Tuple{Int64, Int64}}:
 (1, 1)

julia> interval(P,≥,(2,1),≤,(2,2)) # elements between (2,1) and (2,2)
2-element Vector{Tuple{Int64, Int64}}:
 (2, 1)
 (2, 2)

julia> interval(P,>,(1,1),<,(2,2)) # elements strictly between
2-element Vector{Tuple{Int64, Int64}}:
 (2, 1)
 (1, 2)
julia> interval(P.C,>,1,<,4) # in terms of indices
2-element Vector{Int64}:
 2
 3

A sample of other functions available on posets:

julia> maximal_chains(P)
2-element Vector{Vector{Tuple{Int64, Int64}}}:
 [(1, 1), (2, 1), (2, 2)]
 [(1, 1), (1, 2), (2, 2)]

julia> height(P) # the length of a maximal chain
3

julia> moebiusmatrix(P)
4×4 Matrix{Int64}:
 1  -1  -1   1
 0   1   0  -1
 0   0   1  -1
 0   0   0   1

julia> minima(P)
1-element Vector{Tuple{Int64, Int64}}:
 (1, 1)

julia> maxima(P)
1-element Vector{Tuple{Int64, Int64}}:
 (2, 2)

julia> Q=CPoset(:chain,3)
1<2<3

julia> P1=Poset(Q) # transformed to a Poset with elements 1:3
1<2<3

julia> P⊕ P1 # the ordinal sum
(1, 1)<(2, 1),(1, 2)<(2, 2)<1<2<3

julia> P1*P1
(1, 1)<(2, 1)<(3, 1)<(3, 2)<(3, 3)
(1, 1)<(1, 2)<(2, 2)<(3, 2)
(2, 1)<(2, 2)
(1, 2)<(1, 3)<(2, 3)<(3, 3)
(2, 2)<(2, 3)

julia> P1⊗ P1 # the ordinal product
(1, 1)<(1, 2)<(1, 3)<(2, 1)<(2, 2)<(2, 3)<(3, 1)<(3, 2)<(3, 3)

Finally showpic(p) where p is a CPoset or a Poset gives a graphical display of the poset provided you have the command dot of graphviz installed. It then uses the xdg "open" command to open the resulting .png file. This works on Linux and MacOs but I could not try it on Windows.

see the on-line help on ⊕, ⊗, +, *, chains, chainpoly, covering_chains, coxetermatrix, dual, hasse, height, incidence, induced, interval, isjoinlattice, ismeetlattice, linear_extension, maxima, maximal_chains, minima, moebius, moebiusmatrix, partition, showpic, transitive_closure for more information

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