GAP (Modules) - Chapter 5: Relations

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5 Relations
5.1 Relations: Categories and Representations
5.1-1 IsHomalgRelations
5.1-2 IsHomalgRelationsOfLeftModule
5.1-3 IsHomalgRelationsOfRightModule
5.1-4 IsRelationsOfFinitelyPresentedModuleRep
5.2 Relations: Constructors
5.3 Relations: Properties
5.3-1 CanBeUsedToDecideZeroEffectively
5.3-2 IsInjectivePresentation
5.4 Relations: Attributes
5.5 Relations: Operations and Functions

### 5 Relations

A finite presentation of a module is given by a finite set of generators and a finite set of relations among these generators. In homalg a set of relations of a left/right module is given by a matrix rel, the rows/columns of which are interpreted as relations among n generators, n being the number of columns/rows of the matrix rel.

The data structure of a module in homalg is designed to contain not only one but several sets of relations (together with corresponding sets of generators (--> Chapter 6)). The different sets of relations are linked with so-called transition matrices (--> Chapter 7).

The relations of a homalg module are evaluated in a lazy way. This avoids unnecessary computations.

#### 5.1 Relations: Categories and Representations

##### 5.1-1 IsHomalgRelations

`‣ IsHomalgRelations`( rel ) | ( category )

Returns: `true` or `false`

The GAP category of homalg relations.

##### 5.1-2 IsHomalgRelationsOfLeftModule

`‣ IsHomalgRelationsOfLeftModule`( rel ) | ( category )

Returns: `true` or `false`

The GAP category of homalg relations of a left module.

(It is a subcategory of the GAP category `IsHomalgRelations`.)

##### 5.1-3 IsHomalgRelationsOfRightModule

`‣ IsHomalgRelationsOfRightModule`( rel ) | ( category )

Returns: `true` or `false`

The GAP category of homalg relations of a right module.

(It is a subcategory of the GAP category `IsHomalgRelations`.)

##### 5.1-4 IsRelationsOfFinitelyPresentedModuleRep

`‣ IsRelationsOfFinitelyPresentedModuleRep`( rel ) | ( representation )

Returns: `true` or `false`

The GAP representation of a finite set of relations of a finitely presented homalg module.

(It is a representation of the GAP category `IsHomalgRelations` (5.1-1))

#### 5.2 Relations: Constructors

#### 5.3 Relations: Properties

##### 5.3-1 CanBeUsedToDecideZeroEffectively

`‣ CanBeUsedToDecideZeroEffectively`( rel ) | ( property )

Returns: `true` or `false`

Check if the homalg set of relations rel can be used for normal form reductions.
(no method installed)

##### 5.3-2 IsInjectivePresentation

`‣ IsInjectivePresentation`( rel ) | ( property )

Returns: `true` or `false`

Check if the homalg set of relations rel has zero syzygies.

#### 5.4 Relations: Attributes

#### 5.5 Relations: Operations and Functions

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