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**datatypes
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[1

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[]\OT1/phv/m/n/10 HAPprime \OT1/ptm/m/n/10 is re-leased un-der the GNU Gen-eral
 Pub-lic Li-cense (GPL). This file is part of \OT1/phv/m/n/10 HAP-
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\OT1/phv/m/n/10 prime\OT1/ptm/m/n/10 , though as doc-u-men-ta-tion it is re-lea
sed un-der the GNU Free Doc-u-men-ta-tion Li-cense (see
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[2] (./datatypes.toc
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] [4] [5] [6])
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 [7]
Chapter 1.
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\OT1/ptm/m/n/10.95 The \OT1/phv/m/n/10.95 HAP-prime \OT1/ptm/m/n/10.95 pack-age
 is a \OT1/phv/m/n/10.95 GAP \OT1/ptm/m/n/10.95 pack-age which sup-ple-ments th
e \OT1/phv/m/n/10.95 HAP \OT1/ptm/m/n/10.95 pack-age
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\OT1/ptm/m/n/10.95 ([][]\OT1/pcr/m/n/10.95 http://hamilton.nuigalway.ie/Hap/www
/[][]\OT1/ptm/m/n/10.95 ), pro-vid-ing new and im-proved func-tions for
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[]\OT1/ptm/m/n/10.95 (Chapter [][]6[][]) a free lin-ear ho-mo-mor-phism be-twee
n two $\U/msb/m/n/10.95 F\OML/ptmcm/m/it/10.95 G$\OT1/ptm/m/n/10.95 -
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[8

]
Chapter 2.
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[]\OT1/phv/m/n/10.95 HAPprime \OT1/ptm/m/n/10.95 im-ple-ments the con-struc-tio
n of res-o-lu-tions us-ing this method. The con-struc-
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\OT1/ptm/m/n/10.95 tion is di-vided into two stages. The cre-ation of the first
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/ptm/m/n/10.95 ([][]2.3.2[][]), or
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\OT1/ptm/m/n/10.95 for a res-o-lu-tion of the triv-ial $\U/msb/m/n/10.95 F\OML/
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10.95 , the first two ho-mo-mor-phisms can be stated
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\OT1/ptm/m/n/10.95 with-out cal-cu-la-tion us-ing \OT1/pcr/m/n/10.95 LengthOneR
esolutionPrimePowerGroup \OT1/ptm/m/n/10.95 ([][]2.3.1[][]). Once this ini-
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\OT1/ptm/m/n/10.95 tial se-quence is cre-ated, a longer res-o-lu-tion can be cr
e-ated by re-peated ap-pli-ca-tion of one
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\OT1/ptm/m/n/10.95 of \OT1/pcr/m/n/10.95 ExtendResolutionPrimePowerGroupGF \OT1
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-
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\OT1/ptm/b/n/10.95 GroupGF\OT1/ptm/m/n/10.95 ), \OT1/pcr/m/n/10.95 ExtendResolu
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\OT1/ptm/b/n/10.95 tion-Prime-Pow-er-GroupRad-i-cal\OT1/ptm/m/n/10.95 ) or \OT1
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b/n/10.95 HAPprime:
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\OT1/ptm/b/n/10.95 Ex-ten-dRes-o-lu-tion-Prime-Pow-er-GroupGF2\OT1/ptm/m/n/10.9
5 ), each of which ex-tends the res-o-lu-tion by one
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\OT1/ptm/m/n/10.95 nels of the bound-ary ho-mo-mor-phisms us-ing \OT1/pcr/m/n/1
0.95 KernelOfModuleHomomorphismSplit \OT1/ptm/m/n/10.95 ([][]6.6.3[][])
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\OT1/ptm/m/n/10.95 and finds a min-i-mal set of gen-er-a-tors for this ker-nel 
us-ing \OT1/pcr/m/n/10.95 MinimalGeneratorsModuleGF
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\OT1/ptm/m/n/10.95 ([][]5.5.9[][]). The much faster but memory-hungry \OT1/pcr/
m/n/10.95 ExtendResolutionPrimePowerGroupRadical
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\OT1/ptm/m/n/10.95 ([][]6.6.3[][]) and \OT1/pcr/m/n/10.95 MinimalGeneratorsModu
leRadical \OT1/ptm/m/n/10.95 ([][]5.5.9[][]) re-spec-tively.
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\OT1/pcr/m/n/10.95 ExtendResolutionPrimePowerGroupGF2 \OT1/ptm/m/n/10.95 (\OT1/
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bound-ary ho-
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[10] [11]
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\OT1/ptm/m/n/10.95 We first build each stage ex-plic-itly, start-ing with \OT1/
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or
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[12] [13]
Chapter 3.
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[]\OT1/ptm/m/n/10.95 A list which gives the ba-sis of the al-ge-bra [] (as re-t
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[15]
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[16]
Chapter 4.

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[21] [22]
Chapter 5.
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[23

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[25]
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[]\OT1/ptm/m/n/10.95 Return \OT1/pcr/m/n/10.95 true \OT1/ptm/m/n/10.95 if the m
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[39] [40] [41] [42] [43]
Chapter 6.

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[49] [50] [51]
Chapter 7.

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[]\OT1/pcr/m/n/10.95 "Grlex" \OT1/ptm/m/n/10.95 Graded lex-i-co-graphic or-der-
ing, the equiv-a-lent of \OT1/pcr/m/n/10.95 MonomialGrlexOrdering
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[]\OT1/pcr/m/n/10.95 "Grevlex" \OT1/ptm/m/n/10.95 Graded re-verse lex-i-co-grap
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Chapter 8.
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Chapter 9.
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\OT1/ptm/m/n/10.95 This ex-am-ple com-pares the time taken by \OT1/pcr/m/n/10.9
5 PoincareSeries \OT1/ptm/m/n/10.95 (\OT1/ptm/b/n/10.95 HAP: Poincar-e-Series\O
T1/ptm/m/n/10.95 ) and
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[65] [66]
Chapter 10.
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no-mi-als, each of which
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[70]
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[]\OT1/pcr/m/n/10.95 Best\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Central\OT1/ptm
/m/n/10.95 -\OT1/pcr/m/n/10.95 Subgroup\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 F
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n/10.95 -\OT1/pcr/m/n/10.95 Module\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Homomo
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ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Graded\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 
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[]\OT1/pcr/m/n/10.95 Reverse\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Echelon\OT1/
ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Module\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 
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