Poset(S)

logic.spad line 612 [edit on github]

• S : SetCategory

holds a complete set together with a structure to codify the partial order. for more documentation see: htm

+ : (%, %) -> %

from FiniteGraph(S)

= : (%, %) -> Boolean

from BasicType

addArrow! : (%, NonNegativeInteger, NonNegativeInteger) -> %

addArrow!(s, nm, n1, n2) adds an arrow to the graph s, where: n1 is the index of the start object n2 is the index of the end object This is done in a non-mutable way, that is, the original poset is not changed instead a new one is constructed.

addArrow! : (%, Record(name : String, arrType : NonNegativeInteger, fromOb : NonNegativeInteger, toOb : NonNegativeInteger, xOffset : Integer, yOffset : Integer, map : List(NonNegativeInteger))) -> %

from FiniteGraph(S)

addArrow! : (%, String, S, S) -> %

from FiniteGraph(S)

addArrow! : (%, String, NonNegativeInteger, NonNegativeInteger) -> %

from FiniteGraph(S)

addArrow! : (%, String, NonNegativeInteger, NonNegativeInteger, List(NonNegativeInteger)) -> %

from FiniteGraph(S)

addObject! : (%, S) -> %

addObject!(s, n) adds object with coordinates n to the graph s. This is done in a non-mutable way, that is, the original poset is not changed instead a new one is constructed.

addObject! : (%, Record(value : S, posX : NonNegativeInteger, posY : NonNegativeInteger)) -> %

from FiniteGraph(S)

adjacencyMatrix : % -> Matrix(NonNegativeInteger)

from FiniteGraph(S)

arrowName : (%, NonNegativeInteger, NonNegativeInteger) -> String

from FiniteGraph(S)

arrowsFromArrow : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

arrowsFromNode : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

arrowsToArrow : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

arrowsToNode : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

completeReflexivity : % -> %

Reflexivity requires forall(x): x<=x This function enforces this by making sure that every element has arrow to itself. That is, the leading diagonal is true.

completeTransitivity : % -> %

Transitivity requires forall(x, y, z): x<=y and y<=z implies x<=z This function enforces this by making sure that the composition of any two arrows is also an arrow.

coverMatrix : % -> IncidenceAlgebra(Integer, S)

the covering matrix of a list of elements from a comparison function the list is assumed to be topologically sorted, i.e. w.r. to a linear extension of the comparison function f This function is based on code by Franz Lehner. Notes by Martin Baker on the webpage here:

createWidth : NonNegativeInteger -> NonNegativeInteger

from FiniteGraph(S)

createX : (NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

from FiniteGraph(S)

createY : (NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

from FiniteGraph(S)

cycleClosed : (List(S), String) -> %

from FiniteGraph(S)

cycleOpen : (List(S), String) -> %

from FiniteGraph(S)

deepDiagramSvg : (String, %, Boolean) -> Void

from FiniteGraph(S)

diagramHeight : % -> NonNegativeInteger

from FiniteGraph(S)

diagramSvg : (String, %, Boolean) -> Void

from FiniteGraph(S)

diagramWidth : % -> NonNegativeInteger

from FiniteGraph(S)

diagramsSvg : (String, List(%), Boolean) -> Void

from FiniteGraph(S)

distance : (%, NonNegativeInteger, NonNegativeInteger) -> Integer

from FiniteGraph(S)

distanceMatrix : % -> Matrix(Integer)

from FiniteGraph(S)

finitePoset : (List(S), List(List(Boolean))) -> %

finitePoset(c, s) constructs a finite poset where the set and structure is supplied.

finitePoset : (List(S), Mapping(Boolean, S, S)) -> %

finitePoset(c, p) constructs a finite poset where the set and structure is supplied. The structure is supplied as a predicate function.

flatten : DirectedGraph(%) -> %

from FiniteGraph(S)

getArr : % -> List(List(Boolean))

getArr(s) returns a list of all the arrows (or edges) Note: different from getArrows(s) which is inherited from FiniteGraph(S)

getArrowIndex : (%, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

from FiniteGraph(S)

getArrows : % -> List(Record(name : String, arrType : NonNegativeInteger, fromOb : NonNegativeInteger, toOb : NonNegativeInteger, xOffset : Integer, yOffset : Integer, map : List(NonNegativeInteger)))

from FiniteGraph(S)

getVert : % -> List(S)

getVert(s) returns a list of all the vertices (or objects) of the graph s. Note: different from getVertices(s) which is inherited from FiniteGraph(S)

getVertexIndex : (%, S) -> NonNegativeInteger

from FiniteGraph(S)

getVertices : % -> List(Record(value : S, posX : NonNegativeInteger, posY : NonNegativeInteger))

from FiniteGraph(S)

glb : (%, List(NonNegativeInteger)) -> Union(NonNegativeInteger, "failed")

glb(s, l) 'greatest lower bound' or 'infimum' of l. In this version of glb nodes are represented as index values. Not every subset of a poset will have a glb in which case "failed" will be returned as an error indication.

implies : (%, NonNegativeInteger, NonNegativeInteger) -> Boolean

inDegree : (%, NonNegativeInteger) -> NonNegativeInteger

from FiniteGraph(S)

incidenceMatrix : % -> Matrix(Integer)

from FiniteGraph(S)

indexToObject : (%, NonNegativeInteger) -> S

indexToObject returns the object at a given index.

initial : () -> %

from FiniteGraph(S)

isAcyclic? : % -> Boolean

from FiniteGraph(S)

isAntiChain? : % -> Boolean

isAntiChain?(s) checks if s is an antichain, that is any two elements in s are incomparable.

isAntisymmetric? : % -> Boolean

Antisymmetric requires forall(x, y): x<=y and y<=x iff x=y Returns true if this is the case for every element.

isChain? : % -> Boolean

isChain?(s) checks if s is a chain, that is any two elements in s are comparable.

isDirectSuccessor? : (%, NonNegativeInteger, NonNegativeInteger) -> Boolean

from FiniteGraph(S)

isDirected? : () -> Boolean

from FiniteGraph(S)

isFixPoint? : (%, NonNegativeInteger) -> Boolean

from FiniteGraph(S)

isFunctional? : % -> Boolean

from FiniteGraph(S)

isGreaterThan? : (%, NonNegativeInteger, NonNegativeInteger) -> Boolean

from FiniteGraph(S)

joinIfCan : (%, List(NonNegativeInteger)) -> Union(NonNegativeInteger, "failed")

joinIfCan returns the join of a subset of lattice given by list of elements.

joinIfCan : (%, NonNegativeInteger, NonNegativeInteger) -> Union(NonNegativeInteger, "failed")

joinIfCan(s, a, b) returns the join of 'a' and 'b' In this version of join nodes are represented as index values. In the general case, not every poset will have a join in which case "failed" will be returned as an error indication.

kgraph : (List(S), String) -> %

from FiniteGraph(S)

laplacianMatrix : % -> Matrix(Integer)

from FiniteGraph(S)

latex : % -> String

from SetCategory

le : (%, NonNegativeInteger, NonNegativeInteger) -> Boolean

from Preorder(S)

loopsArrows : % -> List(Loop)

from FiniteGraph(S)

loopsAtNode : (%, NonNegativeInteger) -> List(Loop)

from FiniteGraph(S)

loopsNodes : % -> List(Loop)

from FiniteGraph(S)

looseEquals : (%, %) -> Boolean

from FiniteGraph(S)

lowerSet : % -> %

a subset U with the property that, if x is in U and x >= y, then y is in U

lub : (%, List(NonNegativeInteger)) -> Union(NonNegativeInteger, "failed")

lub(s, l) is 'least upper bound' or 'supremum' of l. In this version of lub nodes are represented as index values. Not every subset of a poset will have a lub in which case "failed" will be returned as an error indication.

map : (%, List(NonNegativeInteger), List(S), Integer, Integer) -> %

from FiniteGraph(S)

mapContra : (%, List(NonNegativeInteger), List(S), Integer, Integer) -> %

from FiniteGraph(S)

max : % -> NonNegativeInteger

from FiniteGraph(S)

max : (%, List(NonNegativeInteger)) -> NonNegativeInteger

from FiniteGraph(S)

meetIfCan : (%, List(NonNegativeInteger)) -> Union(NonNegativeInteger, "failed")

meetIfCan returns the meet of a subset of lattice given by list of elements.

meetIfCan : (%, NonNegativeInteger, NonNegativeInteger) -> Union(NonNegativeInteger, "failed")

meetIfCan(s, a, b) returns the meet of 'a' and 'b' In this version of meet nodes are represented as index values. In the general case, not every poset will have a meet in which case "failed" will be returned as an error indication.

merge : (%, %) -> %

from FiniteGraph(S)

min : % -> NonNegativeInteger

from FiniteGraph(S)

min : (%, List(NonNegativeInteger)) -> NonNegativeInteger

from FiniteGraph(S)

moebius : % -> IncidenceAlgebra(Integer, S)

moebius incidence matrix for this poset This function is based on code by Franz Lehner. Notes by Martin Baker on the webpage here:

nodeFromArrow : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

nodeFromNode : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

nodeToArrow : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

nodeToNode : (%, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

objectToIndex : (%, S) -> NonNegativeInteger

objectToIndex returns the index of a given object.

opposite : % -> %

opposite(s) constructs the opposite in the category theory sense of reversing all the arrows.

outDegree : (%, NonNegativeInteger) -> NonNegativeInteger

from FiniteGraph(S)

powerSetStructure : S -> %

powerSetStructure(set) is a constructor for a Poset where each element is a set (implemented as a list) and with a subset structure. requires S to be a list.

routeArrows : (%, NonNegativeInteger, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

routeNodes : (%, NonNegativeInteger, NonNegativeInteger) -> List(NonNegativeInteger)

from FiniteGraph(S)

setArr : (%, List(List(Boolean))) -> Void

setArr(s, la) sets the list of all arrows (or edges)

setVert : (%, List(S)) -> Void

setVert(s, lv) sets the list of all vertices (or objects)

spanningForestArrow : % -> List(Tree(Integer))

from FiniteGraph(S)

spanningForestNode : % -> List(Tree(Integer))

from FiniteGraph(S)

spanningTreeArrow : (%, NonNegativeInteger) -> Tree(Integer)

from FiniteGraph(S)

spanningTreeNode : (%, NonNegativeInteger) -> Tree(Integer)

from FiniteGraph(S)

subdiagramSvg : (Scene(SCartesian(2)), %, Boolean, Boolean) -> Void

from FiniteGraph(S)

terminal : S -> %

from FiniteGraph(S)

unit : (List(S), String) -> %

from FiniteGraph(S)

upperSet : % -> %

a subset U with the property that, if x is in U and x <= y, then y is in U

zetaMatrix : % -> IncidenceAlgebra(Integer, S)

zetaMatrix(P) returns the matrix of the zeta function This function is based on code by Franz Lehner. Notes by Martin Baker on the webpage here:

~= : (%, %) -> Boolean

from BasicType

SetCategory

FiniteGraph(S)

CoercibleTo(OutputForm)

Preorder(S)

BasicType