WIP: polymake posets

src/Combinatorics/PartiallyOrderedSet/functions.jl

@@ -0,0 +1,224 @@
+function pm_object(P::PartiallyOrderedSet)
+    return P.pm_poset
+end
+
+function partially_ordered_set(pp::Polymake.BigObject)
+  @req startswith(Polymake.type_name(pp), "PartiallyOrderedSet") "Not a polymake PartiallyOrderedSet: $(Polymake.type_name(pp))"
+  return PartiallyOrderedSet(pp)
+end
+
+@doc raw"""
+    partially_ordered_set(covrels::Matrix{Int})
+
+Construct a partially ordered set from covering relations `covrels`, given in
+the form of the adjacency matrix of a Hasse diagram. The covering relations
+must be given in topological order, i.e. `covrels` must be strictly upper
+triangular.
+"""
+function partially_ordered_set(covrels::Matrix{Int})
+  pg = Polymake.Graph{Directed}(IncidenceMatrix(covrels))
+  nelem = nrows(covrels)
+  return partially_ordered_set(Graph{Directed}(pg), 1, nelem)
+end
+
+
+@doc raw"""
+    partially_ordered_set_from_inclusions(i::IncidenceMatrix)
+
+Construct an inclusion based partially ordered set with rows of `i` as co-atoms.
+"""
+function partially_ordered_set_from_inclusions(i::IncidenceMatrix)
+  # FIXME: should not contain extra top node
+  # we want to make sure it is always built from the bottom up
+  # and just need to specify some arbitrary upper bound for the rank
+  pos = Polymake.call_function(:polytope, :lower_hasse_diagram, i, ncols(i))
+  return partially_ordered_set(pos)
+end
+
+
+_pmdec(elem::Int, r::Int) = Polymake.BasicDecoration(Polymake.Set{Int}(elem), r)
+_pmdec(r::Int) = Polymake.BasicDecoration(Polymake.Set{Int}(), r)
+
+@doc raw"""
+    partially_ordered_set(g::Graph{Directed}, minimal_element::Int, maximal_element::Int)
+
+Construct a partially ordered set from a directed graph describing the Hasse diagram.
+The graph must be acyclic and have unique minimal and maximal elements. Edges must be
+directed from bottom to top.
+"""
+function partially_ordered_set(g::Graph{Directed}, minimal_element::Int, maximal_element::Int)
+  # FIXME: do we need min and max to have default values for polymake?
+  stack = [minimal_element]
+  gc = Polymake.Graph{Directed}(pm_object(g))
+  dec = Polymake.NodeMap{Directed, Polymake.BasicDecoration}(gc)
+  # we need to generate some valid rank function
+  es = Polymake.Set{Int}()
+  Polymake._set_entry(dec, minimal_element-1, _pmdec(minimal_element, 0))
+  while !isempty(stack)
+    src = pop!(stack)
+    r = Polymake.decoration_rank(Polymake._get_entry(dec, src-1))
+    for t in neighbors(g, src)
+      rr = Polymake.decoration_rank(Polymake._get_entry(dec, t-1))
+      Polymake._set_entry(dec, t-1, _pmdec(t, max(r+1, rr)))
+      push!(stack, t)
+    end
+  end
+  pos = Polymake.graph.PartiallyOrderedSet{Polymake.BasicDecoration}(
+                                                                     ADJACENCY=gc,
+                                                                     DECORATION=dec,
+                                                                     TOP_NODE=maximal_element-1,
+                                                                     BOTTOM_NODE=minimal_element-1
+                                                                    )
+  return PartiallyOrderedSet(pos)
+end
+
+@doc raw"""
+    partially_ordered_set(g::Graph{Directed}, node_ranks::Dict{Int,Int})
+
+Construct a partially ordered set from a directed graph describing the Hasse diagram.
+The graph must be acyclic and have unique minimal and maximal elements.
+The dictionary `node_ranks` must give a valid rank for each node in the graph, strictly
+increasing from the unique minimal element to the unique maximal element.
+"""
+function partially_ordered_set(g::Graph{Directed}, node_ranks::Dict{Int,Int})
+  # FIXME: do we need min and max to have default values for polymake?
+  gc = Polymake.Graph{Directed}(pm_object(g))
+  dec = Polymake.NodeMap{Directed, Polymake.BasicDecoration}(gc)
+  for n in 1:n_vertices(g)
+    Polymake._set_entry(dec, n-1, _pmdec(n, node_ranks[n]))
+  end
+  bottom = findmin(node_ranks)[2]
+  top = findmax(node_ranks)[2]
+  pos = Polymake.graph.PartiallyOrderedSet{Polymake.BasicDecoration}(
+                                                                     ADJACENCY=gc,
+                                                                     DECORATION=dec,
+                                                                     TOP_NODE=top-1,
+                                                                     BOTTOM_NODE=bottom-1
+                                                                    )
+  return PartiallyOrderedSet(pos)
+end
+
+@doc raw"""
+    comparability_graph(pos::PartiallyOrderedSet)
+
+Return an undirected graph which has an edge for each pair of comparable
+elements in a partially ordered set.
+"""
+comparability_graph(pos::PartiallyOrderedSet) = Graph{Undirected}(pm_object(pos).COMPARABILITY_GRAPH::Polymake.Graph{Undirected})
+
+# this will compute the comparability graph on first use
+# the graph will be kept in the polymake object
+function compare(pos::PartiallyOrderedSet, a::Int, b::Int)
+  cg = comparability_graph(pos)
+  has_edge(cg, a, b) || return false
+  return rank(pos, a) < rank(pos, b)
+end
+
+#=
+ TODO: do we need those?
+function compare(a::PartiallyOrderedSetElement, b::PartiallyOrderedSetElement)
+  @req parent(a) === parent(b) "cannot compare elements in different posets"
+  return compare(parent(a), data(a), data(b))
+end
+
+parent(pose::PartiallyOrderedSetElement) = pose.parent
+data(pose::PartiallyOrderedSetElement) = pose.node_id
+=#
+
+Base.length(p::PartiallyOrderedSet) = pm_object(p).N_NODES::Int
+
+@doc raw"""
+    rank(p::PartiallyOrderedSet, i::Int64)
+
+Return the rank of an element with index `i` in a partially ordered set.
+"""
+function rank(p::PartiallyOrderedSet, i::Int64)
+  dec = pm_object(p).DECORATION::Polymake.NodeMap{Directed,Polymake.BasicDecoration}
+  elemdec = Polymake._get_entry(dec, i-1)::Polymake.BasicDecoration
+  return Polymake.decoration_rank(elemdec)::Int
+end
+
+@doc raw"""
+    maximal_chains(p::PartiallyOrderedSet)
+
+Return the maximal chains of a partially ordered set as a vector of sets.
+The nodes in the chains are not ordered.
+"""
+function maximal_chains(p::PartiallyOrderedSet)
+  mc = Polymake.graph.maximal_chains_of_lattice(pm_object(p))::IncidenceMatrix
+  return row.(Ref(mc), 1:nrows(mc))
+end
+
+function Base.show(io::IO, p::PartiallyOrderedSet)
+  if is_terse(io)
+    print(io, "Partially ordered set")
+  else
+    print(io, "Partially ordered set on $(length(p)) elements")
+  end
+end
+
+function visualize(p::PartiallyOrderedSet; kwargs...)
+  Polymake.visual(pm_object(p); kwargs...)
+end
+
+@doc raw"""
+    face_lattice(p::Union{Polyhedron,Cone,PolyhedralFan,PolyhedralComplex})
+
+Return a partially ordered set encoding the inclusion relation of the faces of `p`.
+For `PolyhedralComplex`, `PolyhedralFan`, and `SimplicialComplex` this contains an
+artificial top node which covers all maximal cells.
+"""
+function face_lattice(p::Union{Polyhedron,Cone,PolyhedralFan,PolyhedralComplex,SimplicialComplex})
+  return partially_ordered_set(pm_object(p).HASSE_DIAGRAM)
+end
+
+@doc raw"""
+    order_polytope(p::PartiallyOrderedSet)
+
+Return the order polytope of a poset.
+See Stanley, Discr Comput Geom 1 (1986).
+TODO: ref
+"""
+function order_polytope(p::PartiallyOrderedSet)
+  # FIXME: check top/bottom ? and more?
+  return polyhedron(Polymake.polytope.order_polytope(pm_object(p)))
+end
+
+@doc raw"""
+    chain_polytope(p::PartiallyOrderedSet)
+
+Return the chain polytope of a poset.
+See Stanley, Discr Comput Geom 1 (1986).
+TODO: ref
+"""
+function chain_polytope(p::PartiallyOrderedSet)
+  # FIXME: check top/bottom ? and more?
+  return polyhedron(Polymake.polytope.chain_polytope(pm_object(p)))
+end
+
+@doc raw"""
+    maximal_ranked_poset(v::AbstractVector{Int})
+
+Maximal ranked partially ordered set with number of nodes per rank given in
+`v`, from bottom to top and excluding the minimal and maximal elements.
+
+See Ahmad, Fourier, Joswig, arXiv:2309.01626
+TODO: ref
+"""
+function maximal_ranked_poset(v::AbstractVector{Int})
+  return partially_ordered_set(Polymake.graph.maximal_ranked_poset(Polymake.Array{Int}(v)))
+end
+
+@doc raw"""
+    graph(p::PartiallyOrderedSet)
+
+Return the directed graph of covering relations in a partially ordered set.
+"""
+graph(p::PartiallyOrderedSet) = Graph{Directed}(pm_object(p).ADJACENCY)
+
+@doc raw"""
+    node_ranks(p::PartiallyOrderedSet)
+
+Return a dictionary mapping each node index to its rank.
+"""
+node_ranks(p::PartiallyOrderedSet) = Dict((i => rank(p, i)) for i in 1:length(p))

src/Combinatorics/PartiallyOrderedSet/structs.jl

@@ -0,0 +1,14 @@
+abstract type AbstractPartiallyOrderedSet end
+
+struct PartiallyOrderedSet <: AbstractPartiallyOrderedSet
+  pm_poset::Polymake.BigObject
+end
+
+#=
+do we need this?
+struct PartiallyOrderedSetElement{T}
+  parent::PartiallyOrderedSet
+  node_id::Int
+  elem::T
+end
+=#

src/Oscar.jl

@@ -275,6 +275,8 @@ include("Combinatorics/SimplicialComplexes.jl")
 include("Combinatorics/OrderedMultiIndex.jl")
 include("Combinatorics/Matroids/JMatroids.jl")
 include("Combinatorics/EnumerativeCombinatorics/EnumerativeCombinatorics.jl")
+include("Combinatorics/PartiallyOrderedSet/structs.jl")
+include("Combinatorics/PartiallyOrderedSet/functions.jl")
 
 include("PolyhedralGeometry/visualization.jl") # needs SimplicialComplex
 

src/exports.jl

@@ -327,6 +327,7 @@ export cellular_primary_decomposition
 export center, has_center, set_center
 export centralizer
 export chain_complex
+export chain_polytope
 export chain_range
 export chamber
 export character_field
@@ -382,6 +383,7 @@ export comm
 export comm!
 export common_components
 export common_refinement
+export comparability_graph
 export complement
 export complement_classes
 export complement_equation
@@ -546,6 +548,7 @@ export exterior_derivative
 export exterior_power
 export f_vector
 export face_fan
+export face_lattice
 export faces
 export facet_degrees
 export facet_indices
@@ -1035,11 +1038,13 @@ export maxes
 export maximal_abelian_quotient, has_maximal_abelian_quotient, set_maximal_abelian_quotient
 export maximal_blocks
 export maximal_cells
+export maximal_chains
 export maximal_cones
 export maximal_extension
 export maximal_groebner_cone
 export maximal_normal_subgroups, has_maximal_normal_subgroups, set_maximal_normal_subgroups
 export maximal_polyhedra, maximal_polyhedra_and_multiplicities
+export maximal_ranked_poset
 export maximal_subgroup_classes, has_maximal_subgroup_classes, set_maximal_subgroup_classes
 export maximal_subgroups
 export metadata
@@ -1179,6 +1184,7 @@ export orbits
 export order, has_order, set_order
 export order_field_of_definition
 export order_of_vanishing
+export order_polytope
 export ordering
 export orders_centralizers
 export orders_class_representatives
@@ -1194,6 +1200,7 @@ export parallel_extension
 export parametrization
 export parametrization_conic
 export parent
+export partially_ordered_set
 export partition
 export partitions
 export patches