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G Overview of the homalg Package Source Code
G.1 Rings, Ring Maps, Matrices, Relations
G.2 The Low Level Algorithms
G.3 Logical Implications for homalg Objects
G.4 The subpackage ResidueClassRingForHomalg
G.5 The homalgTable for GAP4 built-in rings

### G Overview of the homalg Package Source Code

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

Table: The homalg package files
Filename `.gd`/`.gi` | Content
`homalg` | definitions of the basic GAP4 categories
| and some tool functions (e.g. `homalgMode`)
`homalgTable` | dictionaries between homalg
| and the computing engines
`HomalgRing` | internal and external rings
`HomalgRingMap` | ring maps
`HomalgMatrix` | internal and external matrices
`HomalgRelations` | a set of relations

#### G.2 The Low Level Algorithms

In the following CAS or CASystem mean computer algebra systems.

Table: The homalg package files (continued)
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)

#### G.3 Logical Implications for homalg Objects

Table: The homalg package files (continued)
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

#### G.4 The subpackage ResidueClassRingForHomalg

Table: The homalg package files (continued)
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

#### G.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.

Table: The homalg package files (continued)
Filename .gi | Content
`Integers` | the homalgTable for the ring of integers

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