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### 2 Cones and semigroups

#### 2.1 Cones

This section introduces the toric commands which deal with cones and related combinatorial-geometric objects. Recall, a is a strongly convex polyhedral cone ([Ful93], page 4).

##### 2.1-1 InsideCone
 ‣ InsideCone( v, L ) ( function )

This command returns true if the vector v belongs to the interior of the (strongly convex polyhedral) cone generated by the vectors in L.

This procedure does not check if L generates a strongly convex polyhedral cone.

gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
gap> InsideCone(v,L);
false
gap> L:=[[1,0],[3,4]];;
gap> v:=[1,-7]; InsideCone(v,L);
[ 1, -7 ]
false
gap> v:=[4,-3]; InsideCone(v,L);
[ 4, -3 ]
false
gap> v:=[4,-4]; InsideCone(v,L);
[ 4, -4 ]
false
gap> v:=[4,1]; InsideCone(v,L);
[ 4, 1 ]
true


##### 2.1-2 InDualCone
 ‣ InDualCone( v, L ) ( function )

This command returns true if v belongs to the dual of the cone generated by the vectors in L.

gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
gap> InDualCone(v,L);
true
gap> L:=[[1,0],[3,4]];
[ [ 1, 0 ], [ 3, 4 ] ]
gap> v:=[1,-7]; InDualCone(v,L);
[ 1, -7 ]
false
gap> v:=[4,-3]; InDualCone(v,L);
[ 4, -3 ]
true
gap> v:=[4,-4]; InDualCone(v,L);
[ 4, -4 ]
false
gap> v:=[4,1]; InDualCone(v,L);
[ 4, 1 ]
true


##### 2.1-3 PolytopeLatticePoints
 ‣ PolytopeLatticePoints( A, Perps ) ( function )

Input: Perps$$=[v_1,...,v_k]$$ is the list of inward normal" vectors perpendicular to the walls of a polytope $$P$$ in the vector space $$L_0^*\otimes Q$$,
A$$=[a_1,...,a_k]$$ is a k-tuple of integers, where $$a_i$$ denotes the amount the i-th wall" (defined by the normal $$v_i$$) is shifted from the origin (each $$a_i$$ is assumed non-negative).
For example, the polytope $$P$$ with faces [x=0, x=a, y=0, y=b] has Perps$$=[[1,0],[-1,0],[0,1],[0,-1]]$$ and A$$=[0,a,0,b]$$.
Output: the list of points in $$P \cap L_0^*$$.

gap> Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
[ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ]
gap> A:=[0,4,0,3];
[ 0, 4, 0, 3 ]
gap> PolytopeLatticePoints(A,Perps);
[ [ 0, 0 ], [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ],
[ 1, 3 ], [ 2, 0 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 1 ],
[ 3, 2 ], [ 3, 3 ], [ 4, 0 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ] ]
gap> Length(last);
20


##### 2.1-4 Faces
 ‣ Faces( Rays ) ( function )

Input: Rays is a list of rays for the fan $$\Delta$$
Output: All the normals to the faces (hyperplanes of the cone).

gap> Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> Faces(Cones1[1]);
[ [ 1/2, 1 ], [ 2, 1 ] ]
gap> Faces(Cones1[2]);
[ [ -2, -1 ], [ -1, 1 ] ]
gap> Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
gap> Faces(Cones2[1]);
[ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ]
gap> Faces(Cones2[2]);
[ [ 1/3, 5/6, 1 ], [ 1/2, 0, -1 ], [ 2, 0, 1 ] ]


##### 2.1-5 ConesOfFan
 ‣ ConesOfFan( Delta, k ) ( function )

Input: Delta is the fan of cones,
k is the dimension of the cones desired.
Output: The k-dimensional cones in the fan.

##### 2.1-6 NumberOfConesOfFan
 ‣ NumberOfConesOfFan( Delta, k ) ( function )

Input: Delta is the fan of cones in $$V=Q^n$$,
k is the dimension of the cones counted.
Output: The number of k-dimensional cones in the fan.

Idea: The fan Delta is represented as a set of maximal cones. For each maximal cone, look at the k-dimensional faces obtained by taking $$n$$ choose k subsets of the rays describing the cone. Certain of these k-subsets yield the desired cones.

gap> Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
gap> NumberOfConesOfFan(Delta0,2);
6
gap> ConesOfFan(Delta0,2);
[ [ [ 0, 0, 2 ], [ 0, 2, 0 ] ], [ [ 0, 0, 2 ], [ 2, 0, 0 ] ],
[ [ 0, 2, 0 ], [ 1, 2, -2 ] ], [ [ 0, 2, 0 ], [ 2, -2, 1 ] ],
[ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 1, 2, -2 ], [ 2, -2, 1 ] ] ]
gap> ConesOfFan(Delta0,1);
[ [ [ 0, 0, 2 ] ], [ [ 0, 2, 0 ] ], [ [ 1, 2, -2 ] ],
[ [ 2, -2, 1 ] ], [ [ 2, 0, 0 ] ] ]
gap> NumberOfConesOfFan(Delta0,1);
5


##### 2.1-7 ToricStar
 ‣ ToricStar( sigma, Delta ) ( function )

Input: sigma is a cone in the fan, represented by its set of maximal (i.e., highest dimensional) cones.
Delta is the fan of cones in $$V=Q^n$$.
Output: The star of the cone sigma in Delta, i.e., the cones $$\tau$$ which have sigma as a face.

gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ],
>                 [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
gap> #this is the set of maximal cones in the fan Delta
gap> ToricStar([[1,0]],MaxCones);
[  ]
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
[ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 2, -2, 1 ], [ 1, 2, -2 ] ] ]
gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
[ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 1, 1, -2 ] ] ]
gap> ToricStar([[1,0]],MaxCones);
[  ]


#### 2.2 Semigroups

##### 2.2-1 DualSemigroupGenerators
 ‣ DualSemigroupGenerators( L ) ( function )

Input: L is a list of integral $$n$$-vectors generating a cone $$\sigma$$.
Output: the generators of $$S_\sigma$$,

Idea: let $$M$$ be the maximum of the absolute values of the coordinates of the L[i]'s, for each vector $$v$$ in $$[1..M]^n$$, test if $$v$$ is in the dual cone $$\sigma^*$$. If so, add $$v$$ to list of possible generators. Once this for loop is finished, one can check this list for redundant generators. The trick is to simply omit those elements which are of the form $$d_1+d_2$$, where $$d_1$$ and $$d_2$$ are small" elements in the integral dual cone.

This program is not very efficient and should not be used in large examples'' involving semigroups with many'' generators. For example, if you take L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]; then DualSemigroupGenerators(L); can exhaust GAP's memory allocation.

gap> L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
[ [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], [ 2, -1 ], [ 3, -2 ], [ 4, -3 ] ]
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
gap> DualSemigroupGenerators(L);
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, -1, 0 ], [ 1, 0, -1 ] ]

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