Poset S

logic.spad line 613 [edit on github]

holds a complete set together with a structure to codify the partial order. for more documentation see: http://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/index.htm Date Created: Aug 2015 Basic Operations: Related packages: UserDefinedPartialOrdering in setorder.spad Related categories: PartialOrder in catdef.spad Related Domains: DirectedGraph in graph.spad Also See: AMS Classifications:

+: (%, %) -> %

from FiniteGraph S

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> 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, NonNegativeInteger, NonNegativeInteger) -> %

from FiniteGraph S

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

from FiniteGraph S

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

from FiniteGraph S

addObject!: (%, Record(value: S, posX: NonNegativeInteger, posY: 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.

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: url{http://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/moebius/}

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

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

from FiniteGraph S

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

from FiniteGraph S

diagramWidth: % -> NonNegativeInteger

from FiniteGraph S

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

from FiniteGraph S

distanceMatrix: % -> Matrix Integer

from FiniteGraph S

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

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

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

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

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

incidenceMatrix: % -> Matrix Integer

from FiniteGraph S

inDegree: (%, NonNegativeInteger) -> NonNegativeInteger

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.

isDirected?: () -> Boolean

from FiniteGraph S

isDirectSuccessor?: (%, NonNegativeInteger, NonNegativeInteger) -> 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: url{http://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/moebius/}

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: url{http://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/moebius/}

BasicType

CoercibleTo OutputForm

FiniteGraph S

Preorder S

SetCategory