‣ GroebnerBasis( C, rels ) | ( operation ) |
Returns: a dense list
The input is a path category C and a list rels consisting of pairs of parallel morphisms in C. The output is the Groebner basis of rels.
‣ ReducedGroebnerBasisWithGivenGroebnerBasis( C, groebner_basis ) | ( operation ) |
Returns: a dense list
The input is a path category C and a Groebner basis groebner_basis consisting of pairs of parallel morphisms in C. The output is the reduced Groebner basis of groebner_basis.
‣ ReducedGroebnerBasis( C, rels ) | ( operation ) |
Returns: a dense list
The input is a path category C and a list rels consisting of pairs of parallel morphisms in C. The output is the reduced Groebner basis of rels.
‣ IsDescendingForMorphisms( C, f, g, admissible_order ) | ( operation ) |
Returns: a boolean
The input is a path category C, two parallel morphisms f, g and a string admissible_order which takes one of the following two values "Dp" or "dp". The output is whether f \(\succ\) g with respect to the specified admissible order. "Dp" stands for the left-length-lexicographic order under which f \(\succ_{\mathtt{Dp}}\) g if MorphismLength(f) \(\succ\) MorphismLength(g); or MorphismLength(f)\(=\)MorphismLength(g) and the first nonzero entry in MorphismIndices(f)\(-\)MorphismIndices(g) is negative. "dp" stands for the right-length-lexicographic order under which f \(\succ_{\mathtt{dp}}\) g if MorphismLength(f) \(\succ\) MorphismLength(g); or MorphismLength(f)\(=\)MorphismLength(g) and the last nonzero entry in MorphismIndices(f)\(-\)MorphismIndices(g) is negative. For example, if the quiver of C is defined by the string "(*)[x:*->*,y:*->*]", then
\(x^2 \succ_{\mathtt{Dp}} xy \succ_{\mathtt{Dp}} yx \succ_{\mathtt{Dp}} y^2 \succ_{\mathtt{Dp}} x \succ_{\mathtt{Dp}} y \succ_{\mathtt{Dp}} \mathtt{id(*)}\)
\(x^2 \succ_{\mathtt{dp}} yx \succ_{\mathtt{dp}} xy \succ_{\mathtt{dp}} y^2 \succ_{\mathtt{dp}} x \succ_{\mathtt{dp}} y \succ_{\mathtt{dp}} \mathtt{id(*)}\)
‣ IsAscendingForMorphisms( C, f, g, admissible_order ) | ( operation ) |
Returns: a boolean
The input is a path category C, two parallel morphisms f, g and a string admissible_order which takes one of the following two values "Dp" or "dp". The output is whether f \(\prec\) g with respect to the specified admissible order.
‣ IsPathCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of path categories.
‣ IsPathCategoryObject( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in path categories.
‣ IsPathCategoryMorphism( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in path categories.
‣ PathCategory( q ) | ( operation ) |
Returns: a CAP category
The input is a CAP quiver q. The output is the path category of q, i.e., the category whose objects are the objects of q and whose morphisms are lists of morphisms in q, in which the target of any morphism is equal to the source of the next morphism.
‣ ObjectConstructor( C, i ) | ( operation ) |
Returns: a CAP category object
The input is a path category C of CAP quiver \(q\) and a positive integer i. The output is the i'th object in C.
‣ MorphismConstructor( C, s, l, support, t ) | ( operation ) |
Returns: a CAP category morphism
The input is a path category C of CAP quiver \(q\), two objects s, t a nonnegative integer length and a list support of length l consisting of morphisms in \(q\) where the target of each morphism is equal to the source of the next morphism. The output is the morphism in C whose length is l and whose support is support.
‣ AssignSetOfObjects( C[, str] ) | ( operation ) |
Returns: nothing
Assigns the objects of C to global variables. Names of the variables are the concatenation of str with the labels of the objects. The default value of str is the empty string.
‣ AssignSetOfGeneratingMorphisms( C[, str] ) | ( operation ) |
Returns: nothing
Assigns the generating morphisms of C to global variables. Names of the variables are the concatenation of str with the labels of the generating morphisms. The default value of str is the empty string.
‣ OppositePathCategory( C ) | ( attribute ) |
Returns: a path category
Returns the opposite category of the path category C.
‣ CategoryFromNerveData( C ) | ( attribute ) |
‣ UnderlyingQuiver( C ) | ( attribute ) |
Returns: a list of CAP category objects
Returns the defining quiver of the path category C.
‣ ObjectIndex( v ) | ( attribute ) |
Returns: a positive integer
Returns the index of the object v.
‣ ObjectLabel( v ) | ( attribute ) |
Returns: a string
Returns the label of the object v.
‣ MorphismLength( alpha ) | ( attribute ) |
Returns: a positive integer
Returns the length of alpha.
‣ MorphismSupport( alpha ) | ( attribute ) |
Returns: a dense-list of CAP quiver morphisms
Returns the list of morphisms in the underlying quiver which defines alpha.
‣ MorphismIndices( alpha ) | ( attribute ) |
Returns: a dense-list of positive integers
Returns the indices of the morphisms in MorphismSupport(alpha).
‣ MorphismLabel( alpha ) | ( attribute ) |
Returns: a positive integer
Returns the label of the morphism alpha.
‣ IsFinitePathCategory( C ) | ( property ) |
Returns: a boolean
Returns whether C can be enriched over the category of finite sets.
‣ HasFiniteNumberOfMacaulayMorphisms( C, mors ) | ( operation ) |
Returns: a boolean
The input is a path category C and a list of morphisms mors in C (mors is usually the leading terms of a Groebner basis). The output is whether almost all morphisms of C are multiples of elements in mors. In other words, whether the number of morphisms that are not multiples of elements in mors (i.e., Macaulay monomials) is finite.
‣ MacaulayMorphisms( C, mors ) | ( operation ) |
Returns: a boolean
The input is a path category C and a list of morphisms mors in C (mors is usually the leading terms of a Groebner basis). The output is the Macaulay monomials with respect to mors.
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