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### F Overview of the MatricesForHomalg Package Source Code

#### F.1 Rings, Ring Maps, Matrices, Ring Relations

 Filename .gd/.gi Content homalg definitions of the basic GAP4 categories and some tool functions (e.g. homalgMode) homalgTable dictionaries between MatricesForHomalg and the computing engines HomalgRing internal and external rings HomalgRingMap ring maps HomalgMatrix internal and external matrices HomalgRingRelations a set of ring relations

#### F.2 The Low Level Algorithms

In the following CAS or CASystem mean computer algebra systems.

 Filename .gd/.gi Content Tools the elementary matrix operations that can be overwritten using the homalgTable (and hence delegable even to other CASystems) Service the three operations: basis, reduction, and syzygies; they can also be overwritten using the homalgTable (and hence delegable even to other CASystems) Basic higher level operations for matrices (cannot be overwritten using the homalgTable)

#### F.3 Logical Implications for MatricesForHomalg Objects

 Filename .gd/.gi Content LIRNG logical implications for rings LIMAP logical implications for ring maps LIMAT logical implications for matrices COLEM clever operations for lazy evaluated matrices

#### F.4 The subpackage ResidueClassRingForHomalg

 Filename .gd/.gi Content ResidueClassRingForHomalg some global variables ResidueClassRing residue class rings, their elements, and matrices, together with their constructors and operations ResidueClassRingTools the elementary matrix operations for matrices over residue class rings ResidueClassRingBasic the three operations: basis, reduction, and syzygies for matrices over residue class rings

#### F.5 The homalgTable for GAP4 built-in rings

For the purposes of homalg, the ring of integers is, at least up till now, the only ring which is properly supported in GAP4. The GAP4 built-in cababilities for polynomial rings (also univariate) and group rings do not statisfy the minimum requirements of homalg. The GAP4 package Gauss enables GAP to fullfil the homalg requirements for prime fields, and ℤ / p^n.

 Filename .gi Content Integers the homalgTable for the ring of integers

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