A poset can be viewed as a finite \(T_0\)-topological space whose open sets are the downsets. The topological aspects of posets can be found in [Bar11, Section 1.1]
Given a finite poset \(X\), its order complex \(\mathcal{K}(X)\) is the simplicial complex whose vertices are the elements of \(X\) and whose simplices are the nonempty totally ordered subsets of \(X\). If \(K\) is a finite simplicial complex, then its face poset \(\mathcal{X}(K)\) is the finite poset whose elements are the simplices of \(K\) and the order given by inclusion. Moreover, \(X\) (viewed as a finite topological space) and \(\mathcal{K}(X)\) are weak equivalent. Similarly, \(K\) and \(\mathcal{X}(K)\) are weak equivalent.
‣ FacePoset( K ) | ( operation ) |
Returns the face poset of the simplicial complex K.
gap> P:=PosetByCoveringRelations([1,2,3,4], [[2,1], [3,1], [4,2], [4,3]]); <finite poset of size 4> gap> K:=OrderComplex(P); Simplicial complex of dimension 2. gap> PP:=FacePoset(K); <finite poset of size 11> gap> Set(PP); [ [ 1 ], [ 1, 2 ], [ 1, 2, 4 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1, 4 ], [ 2 ], [ 2, 4 ], [ 3 ], [ 3, 4 ], [ 4 ] ]
‣ OrderComplex( X ) | ( operation ) |
Returns the order complex of the finite poset X.
gap> P:=PosetByCoveringRelations([1,2,3,4], [[2,1], [3,1], [4,2], [4,3]]); <finite poset of size 4> gap> K:=OrderComplex(P); Simplicial complex of dimension 2. gap> K!.vertices; [ 1, 2, 3, 4 ] gap> g:=K!.simplices; function( n, i ) ... end gap> g(1,1); [ 1, 2 ] gap> g(1,2); [ 1, 3 ]
‣ FacePoset( f ) | ( operation ) |
The map \(\mathcal{X}(f)\) induced by the simplicial map f in the face posets.
gap> G:=SmallGroup(72,40); <pc group of size 72 with 5 generators> gap> SpG:=BrownPoset(G,2); <finite poset of size 57> gap> Q:=Core(SpG); <finite poset of size 21> gap> K:=OrderComplex(Q); Simplicial complex of dimension 1. gap> L:=OrderComplex(SpG); Simplicial complex of dimension 2. gap> inc:=NaturalMaps(Q)[1]; <order preserving map> gap> Sdinc:=FacePoset(OrderComplex(inc)); <order preserving map>
‣ OrderComplex( f ) | ( operation ) |
The map \(\mathcal{K}(f)\) induced by the poset map f in the order complexes.
gap> G:=SmallGroup(72,40); <pc group of size 72 with 5 generators> gap> SpG:=BrownPoset(G,2); <finite poset of size 57> gap> Q:=Core(SpG); <finite poset of size 21> gap> K:=OrderComplex(Q); Simplicial complex of dimension 1. gap> L:=OrderComplex(SpG); Simplicial complex of dimension 2. gap> ret:=NaturalMaps(Q)[2]; <order preserving map> gap> Kret:=OrderComplex(ret); Simplicial Map
‣ BarycentricSubdivision( X ) | ( operation ) |
The barycentric subdvidision of the poset X, which is the poset \(\mathcal{X}(\mathcal{K}(X))\).
gap> W:=TheWallet(); <finite poset of size 11> gap> SdW:=BarycentricSubdivision(W); <finite poset of size 53> gap> SdW = FacePoset(OrderComplex(W)); true
‣ BarycentricSubdivision( K ) | ( operation ) |
The barycentric subdvidision of the simplicial complex K, which is the simplicial complex \(\mathcal{K}(\mathcal{X}(K))\).
gap> W:=TheWallet(); <finite poset of size 11> gap> K:=OrderComplex(W); Simplicial complex of dimension 2. gap> SdK:=BarycentricSubdivision(K); Simplicial complex of dimension 2.
‣ BarycentricSubdivision( f ) | ( operation ) |
The induced map between the barycentric subdivisions of the simplicial complexes.
‣ BarycentricSubdivision( f ) | ( operation ) |
The induced map between the barycentric subdivisions of the posets.
gap> G:=SmallGroup(72,40); <pc group of size 72 with 5 generators> gap> SpG:=BrownPoset(G,2); <finite poset of size 57> gap> Q:=Core(SpG); <finite poset of size 21> gap> inc:=NaturalMaps(Q)[1]; <order preserving map> gap> SdSpG:=BarycentricSubdivision(SpG); <finite poset of size 255> gap> Sdinc:=BarycentricSubdivision(inc); <order preserving map> gap> TargetMap(Sdinc) = SdSpG; true
‣ PosetHomology( X ) | ( operation ) |
Computes the homology \(H_*(X)\) of the poset X viewed as a topological space.
gap> W:=TheWallet(); <finite poset of size 11> gap> PosetHomology(W); [ [ 0 ], [ ], [ ] ] gap> PosetHomology(MinimalFiniteModelSphere(2)); [ [ 0 ], [ ], [ 0 ] ]
‣ PosetHomology( X, n ) | ( operation ) |
Computes the homology group \(H_{n}(X)\) of the poset X viewed as a topological space.
gap> W:=TheWallet(); <finite poset of size 11> gap> PosetHomology(W,1); [ ] gap> PosetHomology(W,0); [ 0 ] gap> PosetHomology(MinimalFiniteModelSphere(2),2); [ 0 ]
‣ PosetHomology( f, n ) | ( operation ) |
The induced map \(H_{n}(f)\) between the homology groups of degree n by the poset map f.
gap> S2:=MinimalFiniteModelSphere(2); <finite poset of size 6> gap> id_S2:=IdentityMap(S2); <order preserving map> gap> PosetHomology(id_S2,2); [ g1 ] -> [ g1 ]
‣ EulerCharacteristic( X ) | ( attribute ) |
Computes the Euler Characteristic of the poset X viewed as a topological space.
gap> EulerCharacteristic(MinimalFiniteModelSphere(3)); 0 gap> EulerCharacteristic(MinimalFiniteModelSphere(4)); 2 gap> EulerCharacteristic(TheWallet()); 1
‣ FundamentalGroup( X ) | ( operation ) |
Computes the fundamental group of the poset X as is the fundamental group of its order complex.
gap> FundamentalGroup(TheWallet()); <fp group on the generators [ ]> gap> Order(last); 1 gap> F:=FundamentalGroup(MinimalFiniteModelSphere(1)); <fp group of size infinity on the generators [ f1 ]> gap> RelatorsOfFpGroup(F); [ ] gap> GeneratorsOfGroup(F); [ f1 ]
‣ FixedPointsPosetHomomorphism( f ) | ( attribute ) |
Returns the subposet of fixed points of f. We only check names of points, Source(f) is not neccesarily equal to Target(f).
gap> P:=PosetByCoveringRelations([1..5], [[5,3],[5,2],[4,3],[4,2],[3,1],[2,1]]); <finite poset of size 5> gap> A:=SubPoset(P,[1,2,3]); <finite poset of size 3> gap> N:=NaturalMaps(A); [ <order preserving map> ] gap> inc:=N[1]; <order preserving map> gap> F:=FixedPointsPosetHomomorphism(inc); <finite poset of size 3> gap> Set(F); [ 1, 2, 3 ]
‣ HasFixedPointProperty( X ) | ( property ) |
Returns: true or false
Returns true if every order preserving map \(f\colon X \to X\) has a fixed point, false otherwise.
gap> HasFixedPointProperty(MinimalFiniteModelSphere(1)); false
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