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7 More reductions
The reference for this chapter is [Fer17].
7.1 QC Reductions
What we call "down QC reduction" is usually called "QC Reduction". What we call "up QC reduction" is usually called "QC op reduction"
7.1-1 IsDownQCReduction
‣ IsDownQCReduction( X, x, y ) | ( operation ) |
7.1-2 IsUpQCReduction
‣ IsUpQCReduction( X, x, y ) | ( operation ) |
7.1-3 IsQCReduction
‣ IsQCReduction( X, x, y ) | ( operation ) |
7.1-4 QCCore
‣ QCCore( X ) | ( operation ) |
7.1-5 QCCores
‣ QCCores( arg ) | ( operation ) |
Not implemented
7.2 Osaki reductions
7.2-1 IsDownOsakiReduction
‣ IsDownOsakiReduction( X, x ) | ( operation ) |
7.2-2 IsUpOsakiReduction
‣ IsUpOsakiReduction( X, x ) | ( operation ) |
7.2-3 DownOsakiReduction
‣ DownOsakiReduction( X, x ) | ( operation ) |
7.2-4 UpOsakiReduction
‣ UpOsakiReduction( X, x ) | ( operation ) |
7.2-5 DownOsakiCore
‣ DownOsakiCore( X ) | ( operation ) |
7.2-6 UpOsakiCore
‣ UpOsakiCore( X ) | ( operation ) |
7.3 Middle reductions
See [Fer17, Section 3.2.2]
7.3-1 IsDownMiddleReduction
‣ IsDownMiddleReduction( X, x, y ) | ( operation ) |
7.3-2 IsUpMiddleReduction
‣ IsUpMiddleReduction( X, x, y ) | ( operation ) |
7.3-3 IsMiddleReduction
‣ IsMiddleReduction( X, x, y ) | ( operation ) |
7.3-4 MiddleReduction
‣ MiddleReduction( X, x, y ) | ( operation ) |
7.3-5 MiddleReductionCore
‣ MiddleReductionCore( X ) | ( operation ) |
7.4 Edge reductions
7.4-1 IsDownEdgeReduction
‣ IsDownEdgeReduction( X, e ) | ( operation ) |
7.4-2 IsUpEdgeReduction
‣ IsUpEdgeReduction( X, e ) | ( operation ) |
7.4-3 IsEdgeReduction
‣ IsEdgeReduction( X, e ) | ( operation ) |
7.4-4 EdgeReduction
‣ EdgeReduction( X, e ) | ( operation ) |
7.4-5 EdgeReductionCore
‣ EdgeReductionCore( X ) | ( operation ) |
7.5 Random reductions
7.5-1 RandomWeakPointReduction
‣ RandomWeakPointReduction( X ) | ( function ) |
Removes a random weak point, if there is any.
gap> A1:=RandomWeakPointReduction(TheWallet());
# Weak point reduction: 3
<finite poset of size 10>
gap> A2:=RandomWeakPointReduction(TheWallet());
# Weak point reduction: 3
<finite poset of size 10>
gap> A3:=RandomWeakPointReduction(TheWallet());
# Weak point reduction: 11
<finite poset of size 10>
gap> A1 = A2;
true
gap> A1 = A3;
false
gap> RandomWeakPointReduction(MinimalFiniteModelSphere(1));
# No weak points.
<finite poset of size 4>
7.5-2 RandomQCReduction
‣ RandomQCReduction( X ) | ( function ) |
Performs a random QC-reduction, if there is any. Recall that if \(x,y\in X\) are two maximal elements such that \(U_x\cup U_y\) is contractible, then there is a QC-reduction from \(X\) to the quotient space \(X/\{x,y\}\). See [Fer17, Definition 2.2.6].
gap> MaximalElements(TheWallet());
[ 1, 2, 3, 4 ]
gap> RandomQCReduction(TheWallet());
# QC reduction: 1 3
<finite poset of size 10>
gap> RandomQCReduction(TheWallet());
# QC reduction: 3 4
<finite poset of size 10>
gap> RandomQCReduction(MinimalFiniteModelSphere(1));
# No QC reductions.
<finite poset of size 4>
7.5-3 RandomMiddleReduction
‣ RandomMiddleReduction( X ) | ( function ) |
Performs a random middle-reduction, if there is any. Recall that if \(x,y\in X\) are neither maximal nor minimal points such that \(U_z\cap U_y = \{*\}\) for every \(z\in F_x - F_y\) and \(U_z\cap U_x = \{*\}\) for every \(z\in F_y - F_x\), then there is a middle-reduction from \(X\) to the quotient space \(X/ \{x,y\}\). See [Fer17, Definition 3.2.5].
gap> RandomMiddleReduction(TheWallet());
# Middle reduction: 5 6
<finite poset of size 10>
gap> RandomMiddleReduction(MinimalFiniteModelSphere(2));
# No middle reduction.
<finite poset of size 6>
7.5-4 RandomEdgeReduction
‣ RandomEdgeReduction( X ) | ( function ) |
Performs a random edge-reduction, if there is any. Recall that if \(e = (x\prec y)\) is an edge in the Hasse diagram of \(X\), with \(y\) a maximal element, and if \(U_b - e\) (the poset obtained from \(U_b\) by removing the covering relation \(e\)) is contractible, then there is an edge-reduction from \(X\) to \(X-e\). See [Fer17, Definition 3.2.10].
gap> RandomEdgeReduction(TheWallet());
# Edge reduction: [ 7, 11 ]
<finite poset of size 11>
gap> RandomEdgeReduction(MinimalFiniteModelSphere(3));
# No edge reduction.
<finite poset of size 8>
7.5-5 RandomOsakiReduction
‣ RandomOsakiReduction( X ) | ( function ) |
Performs a random Osaki-reduction, if there is any. Recall that if \(x_0\in X\) is an element such that \(U_x\cap U_{x_0}\) is either empty or homotopically trivial for all \(x\in X\), then there is a down Osaki-reduction from \(X\) to \(X/U_{x_0}\). Similarly, if \(F_x\cap F_{x_0}\) is either empty or homotopically trivial for all \(x\in X\) then there is an up Osaki-reduction. An Osaki-reduction is either an up or down Osaki-reduction.
gap> RandomOsakiReduction(TheWallet());
# Up Osaki reduction: 5
<finite poset of size 9>
gap> RandomOsakiReduction(FiniteModelProjectivePlane());
# No Osaki reduction.
<finite poset of size 13>
7.5-6 RandomReductionCore
‣ RandomReductionCore( X ) | ( function ) |
Performs the random reductions described above until reaching a poset with no more reductions. It also shows the list of performed reductions.
gap> T:=TheTriangle();
<finite poset of size 19>
gap> WeakCore(T);
<finite poset of size 19>
gap> RandomReductionCore(T);
# No middle reduction.
# No weak points.
# No Osaki reduction.
# No edge reduction.
# QC reduction: 16 17
# Middle reduction: 5 11
# QC reduction: 13 14
# QC reduction: 13 16
# Edge reduction: [ 8, 2 ]
# Middle reduction: 4 10
# Down Osaki reduction: 7
# Up Osaki reduction: 5
# Weak point reduction: 8
# QC reduction: 7 11
# Middle reduction: 4 7
# Middle reduction: 6 9
# Edge reduction: [ 3, 1 ]
# Weak point reduction: 5
# Up Osaki reduction: 5
# Weak point reduction: 4
# Down Osaki reduction: 3
# Weak point reduction: 1
# Up Osaki reduction: 2
# The poset has size 1.
<finite poset of size 1>
7.5-7 RandomReduction
‣ RandomReduction( X ) | ( function ) |
Performs one of the reductions described above, chosen at random.
gap> RandomReduction(TheWallet());
# Weak point reduction: 3
<finite poset of size 10>
gap> RandomReduction(TheWallet());
# Edge reduction: [ 5, 9 ]
<finite poset of size 11>