‣ SubPoset( X, A ) | ( operation ) |
The subposet of X whose underlying set is the list A of elements of X.
gap> A:=PosetByCoveringRelations([1,2,3,4], [[3,1],[3,2], [4,1],[4,2]]); <finite poset of size 4> gap> SubPoset(A,[1,2]); <finite poset of size 2> gap> CoveringRelations(last); [ ]
‣ CoproductOp( X, A ) | ( operation ) |
Given a list of posets, returns the coproduct of these posets in the given order. It also computes the natural inclusion maps.
‣ Coproduct( X_1, X_2, ... ) | ( function ) |
The coproduct of posets. It also computes the natural inclusion maps.
gap> A:=PosetByCoveringRelations([1,2,3,4], [[3,1],[3,2], [4,1],[4,2]]); <finite poset of size 4> gap> B:=PosetByCoveringRelations([1,2],[]); <finite poset of size 2> gap> C:=TheWallet(); <finite poset of size 11> gap> Coproduct(A,B,C); <finite poset of size 17>
‣ Join( X_1, X_2, ... ) | ( function ) |
The join of posets. It also computes the natural inclusion maps.
gap> A:=PosetByCoveringRelations([1,2,3,4], [[3,1],[3,2], [4,1],[4,2]]); <finite poset of size 4> gap> B:=PosetByCoveringRelations([1,2],[]); <finite poset of size 2> gap> SubPoset(A,[1,2]); <finite poset of size 2> gap> CoveringRelations(last); [ ] gap> C:=TheWallet(); <finite poset of size 11> gap> J:=Join(A,B,C); <finite poset of size 17> gap> a:=Set(J)[1];; b:=J[6];; gap> Ordering(J)(b,a); true
‣ ConePoset( X ) | ( operation ) |
Returns the join of X with the singleton poset. Equivalently, it adds a maximum to X.
gap> W:=TheWallet(); <finite poset of size 11> gap> MaximumPoset(W); fail gap> coneW:=ConePoset(W); <finite poset of size 12> gap> Size(MaximalElements(coneW)); 1
‣ SuspensionPoset( X ) | ( operation ) |
The suspension of the poset X. It is the poset obtained by the join of X with the discrete poset on two elements.
gap> P:=EmptyPoset(); <finite poset of size 0> gap> S1P:=SuspensionPoset(P); <finite poset of size 2> gap> IsomorphismPosets(S1P,MinimalFiniteModelSphere(0)); <order preserving map> gap> S2P:=SuspensionPoset(S1P); <finite poset of size 4> gap> IsomorphismPosets(S2P,MinimalFiniteModelSphere(1)); <order preserving map>
‣ OppositePoset( X ) | ( operation ) |
The oppositve poset of X.
gap> P:=PosetByCoveringRelations([1,2,3], [[3,1], [3,2]]); <finite poset of size 3> gap> Popposite:=OppositePoset(P); <finite poset of size 3> gap> IsomorphismPosets(P,Popposite); fail
‣ MappingCylinderPosetHomomorphism( f ) | ( operation ) |
The mapping cylinder of an order preserving map \(f:X\to Y\) is the poset \(Zf\) whose underlying set is the disjoint union of \(X\) and \(Y\). The order in \(Zf\) is given as following: it keeps the same order in \(X\) and in \(Y\), and if \(x\in X\) and \(y\in Y\), then \(x < y\) if \(f(x)\leq y\). This method also computes the following natural maps: the inclusions of \(X\) and \(Y\) in \(Zf\), and the retraction of \(Zf\) onto \(Y\).
gap> A:=PosetByCoveringRelations([1,2,3,4], [[3,1],[3,2], [4,1],[4,2]]); <finite poset of size 4> gap> B:=PosetByCoveringRelations([1,2,3,4], [[4,2], [4,3], [2,1], [3,1]]); <finite poset of size 4> gap> f:=PosetHomomorphismByImages(A,B, [1, 2, 4, 4]); <order preserving map> gap> MappingCylinderPosetHomomorphism(f); <finite poset of size 8> gap> Zf:=last; <finite poset of size 8>
‣ QuotientPoset( X, A ) | ( operation ) |
Returns the quotient set \(X/A\) with the induced order. The subset \(A\) must be convex: if \(x < z < y\) with \(x,y\in A\), then \(z\in A\). See [Bar11, Section 2.7] for further details.
gap> A:=PosetByCoveringRelations([1,2,3,4], [[4,3],[4,2],[3,1],[2,1]]); <finite poset of size 4> gap> B:=[1,4]; [ 1, 4 ] gap> Q:=QuotientPoset(A,B); The second parameter must be convex. fail gap> Q:=QuotientPoset(A,[4,5]); The second parameter must be a list of elements of the first parameter. fail gap> Q:=QuotientPoset(A,[2,3]); <finite poset of size 3>
‣ WedgePosets( [X_1, x_1, ][X_2, x_2, ]... ) | ( operation ) |
Given a list of pairs \([X_i,x_i]\) where \(X_i\) is a poset and \(x_i\in X_i\), it returns the wedge of these posets on the given points. The \(i\)-th natural map is the inclusion \(X_i\hookrightarrow \vee X_i\).
gap> A:=PosetByCoveringRelations([1,2,3,4], [[4,3],[4,2],[3,1],[2,1]]); <finite poset of size 4> gap> W:=WedgePosets([[A,4], [A,1]]); <finite poset of size 7> gap> IsContractible(W); true
‣ HomPosets( X, Y ) | ( operation ) |
The poset X^Y. Recall that this poset consists of the order preserving maps \(f:X\to Y\) with \(f\leq g\) if \(f(x)\leq g(x)\) for all \(x\in X\).
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> S2:=MinimalFiniteModelSphere(2); <finite poset of size 6> gap> P:=HomPosets(S2,S1); <finite poset of size 44> gap> Q:=HomPosets(S1,S2); <finite poset of size 198>
‣ Endomorphisms( X ) | ( attribute ) |
The endomorphism poset of the poset X. Equivalently, the poset X^{X}.
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> Endo:=Endomorphisms(S1); <finite poset of size 36> gap> Endo = HomPosets(S1,S1); true gap> C:=Core(Endo); <finite poset of size 8> gap> Height(C); 1 gap> PosetHomology(C); [ [ 0, 0, 0, 0, 0 ], [ 0 ] ]
‣ AutomorphismGroup( X ) | ( attribute ) |
The automorphism group \(\mathrm{Aut}(X)\) of \(X\).
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> AutomorphismGroup(S1); C2 x C2
‣ Automorphisms( X ) | ( attribute ) |
The list of the poset automorphisms of \(X\). The first element is the identity map.
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> A:=Automorphisms(S1); [ <order preserving map>, <order preserving map>, <order preserving map>, <order preserving map> ] gap> id:=IdentityMap(S1); <order preserving map> gap> id = A[1]; true
‣ NaturalMaps( X ) | ( attribute ) |
Returns the natural maps of the poset X. For example NaturalMaps of a quotient gives the natural projection.
gap> S1:=MinimalFiniteModelSphere(1); <finite poset of size 4> gap> A:=MinimalElements(S1);; gap> Q:=QuotientPoset(S1,A); <finite poset of size 3> gap> NaturalMaps(Q); [ <order preserving map> ] gap> f:=last[1]; <order preserving map> gap> SourceMap(f) = S1; true gap> TargetMap(f) = Q; true
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