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

F Overview of the MatricesForHomalg Package Source Code

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

Table: The MatricesForHomalg package files
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.

Table: The MatricesForHomalg 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)

F.3 Logical Implications for MatricesForHomalg Objects

Table: The MatricesForHomalg 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

F.4 The subpackage ResidueClassRingForHomalg

Table: The MatricesForHomalg 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

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.

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

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