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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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