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