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### 13 Further examples

gap> R := HomalgFieldOfRationalsInSingular( ) * "x,y";
Q[x,y]
gap> ZF := ZariskiFrameOfAffineSpectrum( R );
The frame of Zariski open subsets of the affine spectrum of Q[x,y]
gap> A := OpenSubsetOfSpecByRadicalColumn( HomalgMatrix( "[ x ]", 1, 1, R ) );
D_{Q[x,y]}( <...> )
gap> B := OpenSubsetOfSpec( "[ x^2 ]", R );
D_{Q[x,y]}( <...> )
gap> A = B;
true
gap> C := OpenSubsetOfSpecByRadicalColumn( "[ y ]", R );
D_{Q[x,y]}( <...> )
gap> D := OpenSubsetOfSpec( "[ x^2*y^3 ]", R );
D_{Q[x,y]}( <...> )
gap> A * C = D;
true
gap> T := TerminalObject( ZF );
D_{Q[x,y]}( <...> )
gap> I := InitialObject( ZF );
D_{Q[x,y]}( <...> )

gap> R := HomalgFieldOfRationalsInSingular( ) * "x,y";
Q[x,y]
gap> ZC := ZariskiCoframeOfAffineSpectrum( R );
The coframe of Zariski closed subsets of the affine spectrum of Q[x,y]
gap> A := ClosedSubsetOfSpecByRadicalColumn( HomalgMatrix( "[ x ]", 1, 1, R ) );
V_{Q[x,y]}( <...> )
gap> B := ClosedSubsetOfSpec( "[ x^2 ]", R );
V_{Q[x,y]}( <...> )
gap> A = B;
true
gap> C := ClosedSubsetOfSpecByRadicalColumn( "[ y ]", R );
V_{Q[x,y]}( <...> )
gap> D := ClosedSubsetOfSpec( "[ x^2*y^3 ]", R );
V_{Q[x,y]}( <...> )
gap> A + C = D;
true
gap> T := TerminalObject( ZC );
V_{Q[x,y]}( <...> )
gap> Dimension( T );
2
gap> I := InitialObject( ZC );
V_{Q[x,y]}( <...> )
gap> Dimension( I );
-1
gap> F := DistinguishedLocallyClosedPart( -( A * C ) );
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> Display( F );
V( <> ) \ V( <y> )

gap> Q := HomalgFieldOfRationalsInSingular( );
Q
gap> R := Q * "x,y";
Q[x,y]
gap> I := HomalgMatrix( "y", 1, 1, R );
<A 1 x 1 matrix over an external ring>
gap> x := ClosedSubsetOfSpec( I );
V_{Q[x,y]}( <...> )
gap> Display( x );
V( <y> )
gap> IsClosed( x );
true
gap> Dimension( x );
1
gap> y := ClosedSubsetOfSpec( "x", R );
V_{Q[x,y]}( <...> )
gap> d := ClosedSubsetOfSpec( "x+y-1", R );
V_{Q[x,y]}( <...> )
gap> xuy := x + y;
V_{Q[x,y]}( <...> )
gap> Display( xuy );
{ V( <y> ) ∪ V( <x> ) }
gap> IsClosed( xuy );
true
gap> mxuy := -xuy;
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> Display( mxuy );
V( <> )

\ { V( <y> ) ∪ V( <x> ) }
gap> IsClosed( mxuy );
false
gap> IsOpen( mxuy );
true
gap> xmy := x - y;
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> Display( xmy );
V( <y> )

\ V( <x> )
gap> xmy2 := xmy - y;
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> Display( xmy2 );
V( <y> )

\ V( <x> )
gap> lc := xuy - d;
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> lc0 := lc - 0;
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> IsIdenticalObj( lc, lc0 );
true
gap> IsLocallyClosed( lc );
true
gap> IsClosed( lc );
false
gap> Dimension( lc );
1
gap> tp := d * xuy;
V_{Q[x,y]}( <...> )
gap> Dimension( tp );
0
gap> c := lc + tp;
( V_{Q[x,y]}( I1 ) \ V_{Q[x,y]}( J1 ) ) ∪ ( V_{Q[x,y]}( I2 ) \ V_{Q[x,y]}( J2 ) )
gap> c0 := c - 0;
( V_{Q[x,y]}( I1 ) \ V_{Q[x,y]}( J1 ) ) ∪ ( V_{Q[x,y]}( I2 ) \ V_{Q[x,y]}( J2 ) )
gap> IsIdenticalObj( c, c0 );
true
gap> c[1];
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> c[2];
V_{Q[x,y]}( I ) \ V_{Q[x,y]}( J )
gap> Dimension( c );
1
gap> c = xuy;
true
gap> cc := CanonicalObject( c );
V_{Q[x,y]}( <...> )
gap> cc = xuy;
true
gap> t := c - lc;
( V_{Q[x,y]}( I1 ) \ V_{Q[x,y]}( J1 ) )
gap> Display( t );
( V( <x+y-1,y^2-y,x+y-1,x*y> )

\ ∅ )
gap> IsClosed( t );
true
gap> z := c - c;
( V_{Q[x,y]}( I1 ) \ V_{Q[x,y]}( J1 ) )
gap> Display( z );
( V( <x*y> )

\ { V( <x+y-1,y^2-y> ) ∪ V( <x*y> ) } )
gap> z := StandardizedObject( z );
∅
gap> Display( z );
( ∅

\ ∅ )

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