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7 Relations
7.1 Relations: Categories and Representations
7.1-1 IsHomalgRelations
7.1-2 IsHomalgRelationsOfLeftModule
7.1-3 IsHomalgRelationsOfRightModule
7.1-4 IsRelationsOfFinitelyPresentedModuleRep
7.2 Relations: Constructors
7.3 Relations: Properties
7.3-1 CanBeUsedToDecideZeroEffectively
7.3-2 IsInjectivePresentation
7.4 Relations: Attributes
7.4-1 FreeResolution
7.5 Relations: Operations and Functions

### 7 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 8)). The different sets of relations are linked with so-called transition matrices (--> Chapter 9).

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

#### 7.1 Relations: Categories and Representations

##### 7.1-1 IsHomalgRelations

`> IsHomalgRelations`( rel ) | ( category )

Returns: `true` or `false`

The GAP category of homalg relations.

##### 7.1-2 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`.)

##### 7.1-3 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`.)

##### 7.1-4 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` (7.1-1))

#### 7.2 Relations: Constructors

#### 7.3 Relations: Properties

##### 7.3-1 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)

##### 7.3-2 IsInjectivePresentation

`> IsInjectivePresentation`( rel ) | ( property )

Returns: `true` or `false`

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

#### 7.4 Relations: Attributes

##### 7.4-1 FreeResolution

`> FreeResolution`( rel ) | ( attribute )

Returns: a homalg complex

The computed (part of a) free resolution of the module presented by homalg set of relations rel.

#### 7.5 Relations: Operations and Functions

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