‣ ElementsAbove( X, x ) | ( operation ) |
The subposet F_x=\{y\,:\, y \geq x\}.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> ElementsAbove(P,1); <finite poset of size 2> gap> Set(last); [ 1, 3 ]
‣ ElementsStrictlyAbove( X, x ) | ( operation ) |
The subposet \hat{F}_x=\{y\,:\, y > x\}.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> Set(ElementsStrictlyAbove(P,1)); [ 3 ]
‣ ElementsBelow( X, x ) | ( operation ) |
The subposet U_x=\{y\,:\, y \leq x\}.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> Set(ElementsBelow(P,3)); [ 1, 2, 3 ]
‣ ElementsStrictlyBelow( X, x ) | ( operation ) |
The subposet \hat{U}_x=\{y\,:\, y < x\}.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> ElementsStrictlyBelow(P,3); <finite poset of size 2> gap> Set(last); [ 1, 2 ]
‣ ElementsComparable( X, x ) | ( operation ) |
The subposet given by the elements which are comparable with x.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> ElementsComparable(P,2); <finite poset of size 2> gap> Set(last); [ 2, 3 ]
‣ ElementsStrictlyComparable( X, x ) | ( operation ) |
The subposet given by the elements which are comparable with x and different from x.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> ElementsStrictlyComparable(P,3); <finite poset of size 2> gap> Set(last); [ 1, 2 ]
‣ RemovePoint( X, x ) | ( operation ) |
The subposet X\setminus x.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> Q:=RemovePoint(P,3); <finite poset of size 2> gap> Set(Q); [ 1, 2 ]
‣ RemoveCoveringRelation( X, e ) | ( operation ) |
The subposet obtained by removing the covering relation e from the Hasse diagram of X.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1],[3,2]]); <finite poset of size 3> gap> Q:=RemoveCoveringRelation(P,[3,1]); <finite poset of size 3> gap> CoveringRelations(Q); [ [ 3, 2 ] ]
‣ Grading( X ) | ( attribute ) |
Returns a grading if the poset X is graded and fail otherwise. The grading is chosen so that the minimum grade is 0.
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> g:=Grading(S1); function( x ) ... end gap> Min:=MinimalElements(S1);; gap> a:=Min[1];; gap> g(a); 0 gap> b:=Set(S1)[3];; gap> g(b); 1
‣ Height( X, x ) | ( operation ) |
The Height of a point in a poset.
gap> P:=PosetByCoveringRelations([1..5],[[5,4],[5,3],[4,2],[3,1],[2,1]]); <finite poset of size 5> gap> Grading(P); fail gap> Height(P,5); 3
‣ Height( X ) | ( attribute ) |
The height of a poset.
gap> Height(MinimalFiniteModelSphere(4)); 4 gap> Height(MinimalFiniteModelSphere(5)); 5 gap> Height(TheWallet()); 2
‣ ChainsPoset( X ) | ( operation ) |
The set of chains of X. Each chain is represented as a list of points, ordered according to the order on X, the first element being the maximum of the chain.
gap> P:=PosetByCoveringRelations([1..5],[[5,4],[5,3],[4,2],[3,1],[2,1]]); <finite poset of size 5> gap> ChainsPoset(P); [ [ 1 ], [ 2 ], [ 2, 1 ], [ 3 ], [ 3, 1 ], [ 4 ], [ 4, 1 ], [ 4, 2 ], [ 4, 2, 1 ], [ 5 ], [ 5, 1 ], [ 5, 2 ], [ 5, 2, 1 ], [ 5, 3 ], [ 5, 3, 1 ], [ 5, 4 ], [ 5, 4, 1 ], [ 5, 4, 2 ], [ 5, 4, 2, 1 ] ]
‣ MobiusMatrix( arg ) | ( attribute ) |
‣ MobiusFunction( X ) | ( attribute ) |
The Mobius function \mu of X.
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