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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))

5.3 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.

5.5 Relations: Operations and Functions

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