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### 5 Category of rows

#### 5.1 Supported CAP operations

##### 5.1-1 CategoryOfRows of an arbitrary ring

The following CAP operations are supported:

##### 5.1-2 CategoryOfRows of an exterior algebra over a field

The following additional CAP operations are supported:

##### 5.1-3 CategoryOfRows of a commutative ring

The following additional CAP operations are supported:

##### 5.1-4 CategoryOfRows of a field

The following additional CAP operations are supported:

#### 5.2 GAP Categories

##### 5.2-1 IsCategoryOfRowsObject
 ‣ IsCategoryOfRowsObject( object ) ( filter )

Returns: true or false

The GAP category of objects in the category of rows over a ring R.

#### 5.3 Random Methods in Category of Rows

##### 5.3-1 RandomObjectByList
 ‣ RandomObjectByList( C, L ) ( operation )

Returns: an object in a category of rows

The arguments are a category C and a non-empty list L of non-negative integers. The output is an object whose rank is a random element in L.

##### 5.3-2 RandomObjectByInteger
 ‣ RandomObjectByInteger( C, n ) ( operation )

Returns: an object in a category of rows

The arguments are a category of rows C and a non-negative integer n. The output is an object whose rank is at most n.

##### 5.3-3 RandomMorphismWithFixedSourceAndRangeByList
 ‣ RandomMorphismWithFixedSourceAndRangeByList( S, R, L ) ( operation )

Returns: a morphism in a category of rows

The arguments are two objects S, R and a list L of integers or elements in the underlying ring. The output is a morphism from S to R whose matrix is an L-linear combination of random matrices over the underlying ring.

##### 5.3-4 RandomMorphismWithFixedSourceAndRangeByInteger
 ‣ RandomMorphismWithFixedSourceAndRangeByInteger( S, R, n ) ( operation )

Returns: a morphism in a category of rows

The arguments are two objects S, R and an integer n. The output is a morphism from S to R defined by a random matrix over the underlying ring. Particularly, the argument n will be disregarded.

##### 5.3-5 RandomMorphismWithFixedSourceByList
 ‣ RandomMorphismWithFixedSourceByList( S, L ) ( operation )

Returns: a morphism in a category of rows

The arguments are an object S and a list L consisting of two lists: a non-empty list of non-negative integers and a list of integers or elements in the underlying ring. The output is a morphism from S to an object R whose rank is a random element in L[1]. The matrix of the morphism is an L[2]-linear combination of random matrices over the underlying ring.

##### 5.3-6 RandomMorphismWithFixedSourceByInteger
 ‣ RandomMorphismWithFixedSourceByInteger( S, n ) ( operation )

Returns: a morphism in a category of rows

The arguments are an object S and a non-negative integer n. The output is a morphism from S to an object R whose rank is at most n.

##### 5.3-7 RandomMorphismWithFixedRangeByList
 ‣ RandomMorphismWithFixedRangeByList( R, L ) ( operation )

Returns: a morphism in a category of rows

The arguments are an object R and a list L consisting of two lists: a non-empty list of non-negative integers and a list of integers or elements in the underlying ring. The output is a morphism to R from an object S whose rank is a random element in L[1]. The matrix of the morphism is an L[2]-linear combination of random matrices over the underlying ring.

##### 5.3-8 RandomMorphismWithFixedRangeByInteger
 ‣ RandomMorphismWithFixedRangeByInteger( R, n ) ( operation )

Returns: a morphism in a category of rows

The arguments are an object R and a non-negative integer n. The output is a morphism to R from an object S whose rank is at most n.

##### 5.3-9 RandomMorphismByList
 ‣ RandomMorphismByList( C, L ) ( operation )

Returns: a morphism in a category of rows

The arguments are a category of rows C and a list L consisting of 3 lists: two non-empty lists of non-negative integers and a list of integers or elements in the underlying ring. The output is a morphism from an object S to an object R whose ranks are random elements in L[1] resp. L[2]. Its matrix is an L[3]-linear combination of random matrices over the underlying ring.

##### 5.3-10 RandomMorphismByInteger
 ‣ RandomMorphismByInteger( C, n ) ( operation )

Returns: a morphism in a category of rows

The arguments are a category of rows C and a non-negative integer n. The output is a morphism whose source and range ranks are at most n.

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