CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutSign UpSign In

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place.

| Download

GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

Views: 418346
Tweet
Share
Share
1
  

2
  1 Introduction

3
  

4
  This  package is part of the homalg project [hpa10]. The role of the package

5
  is described in the manual of the homalg package.

6
  

7
  

8
  1.1 Ring Constructions for Supported External Computer Algebra Systems

9
  

10
  Here are some of the supported ring constructions:

11
  

12
  

13
  1.1-1 external GAP

14
  

15
    Example  

16
    gap> ZZ := HomalgRingOfIntegersInExternalGAP( );

17
    Z

18
    gap> Display( ZZ );

19
    <An external ring residing in the CAS GAP>

20
    gap> F2 := HomalgRingOfIntegersInExternalGAP( 2, ZZ );

21
    GF(2)

22
    gap> Display( F2 );

23
    <An external ring residing in the CAS GAP>

24
  

25
  

26
  F2 := HomalgRingOfIntegersInExternalGAP( 2 ) would launch another GAP.

27
  

28
    Example  

29
    gap> Z4 := HomalgRingOfIntegersInExternalGAP( 4, ZZ );

30
    Z/4Z

31
    gap> Display( Z4 );

32
    <An external ring residing in the CAS GAP>

33
    gap> Z_4 := HomalgRingOfIntegersInExternalGAP( ZZ ) / 4;

34
    Z/( 4 )

35
    gap> Display( Z_4 );

36
    <A residue class ring>

37
    gap> Q := HomalgFieldOfRationalsInExternalGAP( ZZ );

38
    Q

39
    gap> Display( Q );

40
    <An external ring residing in the CAS GAP>

41
  

42
  

43
  

44
  1.1-2 Singular

45
  

46
    Example  

47
    gap> F2 := HomalgRingOfIntegersInSingular( 2 );

48
    GF(2)

49
    gap> Display( F2 );

50
    <An external ring residing in the CAS Singular>

51
    gap> F2s := HomalgRingOfIntegersInSingular( 2, "s" ,F2 );

52
    GF(2)(s)

53
    gap> Display( F2s );

54
    <An external ring residing in the CAS Singular>

55
    gap> ZZ := HomalgRingOfIntegersInSingular( F2 );

56
    Z

57
    gap> Display( ZZ );

58
    <An external ring residing in the CAS Singular>

59
    gap> Q := HomalgFieldOfRationalsInSingular( F2 );

60
    Q

61
    gap> Display( Q );

62
    <An external ring residing in the CAS Singular>

63
    gap> Qs := HomalgFieldOfRationalsInSingular( "s", F2 );

64
    Q(s)

65
    gap> Display( Qs );

66
    <An external ring residing in the CAS Singular>

67
    gap> Qi := HomalgFieldOfRationalsInSingular( "i", "i^2+1", Q );

68
    Q[i]/(i^2+1)

69
    gap> Display( Qi );

70
    <An external ring residing in the CAS Singular>

71
  

72
  

73
  Q := HomalgFieldOfRationalsInSingular( ) would launch another Singular.

74
  

75
    Example  

76
    gap> F2xyz := F2 * "x,y,z";

77
    GF(2)[x,y,z]

78
    gap> Display( F2xyz );

79
    <An external ring residing in the CAS Singular>

80
    gap> F2sxyz := F2s * "x,y,z";

81
    GF(2)(s)[x,y,z]

82
    gap> Display( F2sxyz );

83
    <An external ring residing in the CAS Singular>

84
    gap> F2xyzw := F2xyz * "w";

85
    GF(2)[x,y,z][w]

86
    gap> Display( F2xyzw );

87
    <An external ring residing in the CAS Singular>

88
    gap> F2sxyzw := F2sxyz * "w";

89
    GF(2)(s)[x,y,z][w]

90
    gap> Display( F2sxyzw );

91
    <An external ring residing in the CAS Singular>

92
    gap> ZZxyz := ZZ * "x,y,z";

93
    Z[x,y,z]

94
    gap> Display( ZZxyz );

95
    <An external ring residing in the CAS Singular>

96
    gap> ZZxyzw := ZZxyz * "w";

97
    Z[x,y,z][w]

98
    gap> Display( ZZxyzw );

99
    <An external ring residing in the CAS Singular>

100
    gap> Qxyz := Q * "x,y,z";

101
    Q[x,y,z]

102
    gap> Display( Qxyz );

103
    <An external ring residing in the CAS Singular>

104
    gap> Qsxyz := Qs * "x,y,z";

105
    Q(s)[x,y,z]

106
    gap> Display( Qsxyz );

107
    <An external ring residing in the CAS Singular>

108
    gap> Qixyz := Qi * "x,y,z";

109
    (Q[i]/(i^2+1))[x,y,z]

110
    gap> Display( Qixyz );

111
    <An external ring residing in the CAS Singular>

112
    gap> Qxyzw := Qxyz * "w";

113
    Q[x,y,z][w]

114
    gap> Display( Qxyzw );

115
    <An external ring residing in the CAS Singular>

116
    gap> Qsxyzw := Qsxyz * "w";

117
    Q(s)[x,y,z][w]

118
    gap> Display( Qsxyzw );

119
    <An external ring residing in the CAS Singular>

120
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );

121
    Q[x,y,z]<Dx,Dy,Dz>

122
    gap> Display( Dxyz );

123
    <An external ring residing in the CAS Singular>

124
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );

125
    Q{e,f,g}

126
    gap> Display( Exyz );

127
    <An external ring residing in the CAS Singular>

128
    gap> Dsxyz := RingOfDerivations( Qsxyz, "Dx,Dy,Dz" );

129
    Q(s)[x,y,z]<Dx,Dy,Dz>

130
    gap> Display( Dsxyz );

131
    <An external ring residing in the CAS Singular>

132
    gap> Esxyz := ExteriorRing( Qsxyz, "e,f,g" );

133
    Q(s){e,f,g}

134
    gap> Display( Esxyz );

135
    <An external ring residing in the CAS Singular>

136
    gap> Dixyz := RingOfDerivations( Qixyz, "Dx,Dy,Dz" );

137
    (Q[i]/(i^2+1))[x,y,z]<Dx,Dy,Dz>

138
    gap> Display( Dixyz );

139
    <An external ring residing in the CAS Singular>

140
    gap> Eixyz := ExteriorRing( Qixyz, "e,f,g" );

141
    (Q[i]/(i^2+1)){e,f,g}

142
    gap> Display( Eixyz );

143
    <An external ring residing in the CAS Singular>

144
  

145
  

146
  

147
  1.1-3 MAGMA

148
  

149
    Example  

150
    gap> ZZ := HomalgRingOfIntegersInMAGMA( );

151
    Z

152
    gap> Display( ZZ );

153
    <An external ring residing in the CAS MAGMA>

154
    gap> F2 := HomalgRingOfIntegersInMAGMA( 2, ZZ );

155
    GF(2)

156
    gap> Display( F2 );

157
    <An external ring residing in the CAS MAGMA>

158
  

159
  

160
  F2 := HomalgRingOfIntegersInMAGMA( 2 ) would launch another MAGMA.

161
  

162
    Example  

163
    gap> Z_4 := HomalgRingOfIntegersInMAGMA( ZZ ) / 4;

164
    Z/( 4 )

165
    gap> Display( Z_4 );

166
    <A residue class ring>

167
    gap> Q := HomalgFieldOfRationalsInMAGMA( ZZ );

168
    Q

169
    gap> Display( Q );

170
    <An external ring residing in the CAS MAGMA>

171
    gap> F2xyz := F2 * "x,y,z";

172
    GF(2)[x,y,z]

173
    gap> Display( F2xyz );

174
    <An external ring residing in the CAS MAGMA>

175
    gap> Qxyz := Q * "x,y,z";

176
    Q[x,y,z]

177
    gap> Display( Qxyz );

178
    <An external ring residing in the CAS MAGMA>

179
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );

180
    Q{e,f,g}

181
    gap> Display( Exyz );

182
    <An external ring residing in the CAS MAGMA>

183
  

184
  

185
  

186
  1.1-4 Macaulay2

187
  

188
    Example  

189
    gap> ZZ := HomalgRingOfIntegersInMacaulay2( );

190
    Z

191
    gap> Display( ZZ );

192
    <An external ring residing in the CAS Macaulay2>

193
    gap> F2 := HomalgRingOfIntegersInMacaulay2( 2, ZZ );

194
    GF(2)

195
    gap> Display( F2 );

196
    <An external ring residing in the CAS Macaulay2>

197
  

198
  

199
  F2 := HomalgRingOfIntegersInMacaulay2( 2 ) would launch another Macaulay2.

200
  

201
    Example  

202
    gap> Z_4 := HomalgRingOfIntegersInMacaulay2( ZZ ) / 4;

203
    Z/( 4 )

204
    gap> Display( Z_4 );

205
    <A residue class ring>

206
    gap> Q := HomalgFieldOfRationalsInMacaulay2( ZZ );

207
    Q

208
    gap> Display( Q );

209
    <An external ring residing in the CAS Macaulay2>

210
    gap> F2xyz := F2 * "x,y,z";

211
    GF(2)[x,y,z]

212
    gap> Display( F2xyz );

213
    <An external ring residing in the CAS Macaulay2>

214
    gap> Qxyz := Q * "x,y,z";

215
    Q[x,y,z]

216
    gap> Display( Qxyz );

217
    <An external ring residing in the CAS Macaulay2>

218
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );

219
    Q[x,y,z]<Dx,Dy,Dz>

220
    gap> Display( Dxyz );

221
    <An external ring residing in the CAS Macaulay2>

222
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );

223
    Q{e,f,g}

224
    gap> Display( Exyz );

225
    <An external ring residing in the CAS Macaulay2>

226
  

227
  

228
  

229
  1.1-5 Sage

230
  

231
    Example  

232
    gap> ZZ := HomalgRingOfIntegersInSage( );

233
    Z

234
    gap> Display( ZZ );

235
    <An external ring residing in the CAS Sage>

236
    gap> F2 := HomalgRingOfIntegersInSage( 2, ZZ );

237
    GF(2)

238
    gap> Display( F2 );

239
    <An external ring residing in the CAS Sage>

240
  

241
  

242
  F2 := HomalgRingOfIntegersInSage( 2 ) would launch another Sage.

243
  

244
    Example  

245
    gap> Z_4 := HomalgRingOfIntegersInSage( ZZ ) / 4;

246
    Z/( 4 )

247
    gap> Display( Z_4 );

248
    <A residue class ring>

249
    gap> Q := HomalgFieldOfRationalsInSage( ZZ );

250
    Q

251
    gap> Display( Q );

252
    <An external ring residing in the CAS Sage>

253
    gap> F2x := F2 * "x";

254
    GF(2)[x]

255
    gap> Display( F2x );

256
    <An external ring residing in the CAS Sage>

257
    gap> Qx := Q * "x";

258
    Q[x]

259
    gap> Display( Qx );

260
    <An external ring residing in the CAS Sage>

261
  

262
  

263
  

264
  1.1-6 Maple

265
  

266
    Example  

267
    gap> ZZ := HomalgRingOfIntegersInMaple( );

268
    Z

269
    gap> Display( ZZ );

270
    <An external ring residing in the CAS Maple>

271
    gap> F2 := HomalgRingOfIntegersInMaple( 2, ZZ );

272
    GF(2)

273
    gap> Display( F2 );

274
    <An external ring residing in the CAS Maple>

275
  

276
  

277
  F2 := HomalgRingOfIntegersInMaple( 2 ) would launch another Maple.

278
  

279
    Example  

280
    gap> Z4 := HomalgRingOfIntegersInMaple( 4, ZZ );

281
    Z/4Z

282
    gap> Display( Z4 );

283
    <An external ring residing in the CAS Maple>

284
    gap> Z_4 := HomalgRingOfIntegersInMaple( ZZ ) / 4;

285
    Z/( 4 )

286
    gap> Display( Z_4 );

287
    <A residue class ring>

288
    gap> Q := HomalgFieldOfRationalsInMaple( ZZ );

289
    Q

290
    gap> Display( Q );

291
    <An external ring residing in the CAS Maple>

292
    gap> F2xyz := F2 * "x,y,z";

293
    GF(2)[x,y,z]

294
    gap> Display( F2xyz );

295
    <An external ring residing in the CAS Maple>

296
    gap> Qxyz := Q * "x,y,z";

297
    Q[x,y,z]

298
    gap> Display( Qxyz );

299
    <An external ring residing in the CAS Maple>

300
    gap> Dxyz := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );

301
    Q[x,y,z]<Dx,Dy,Dz>

302
    gap> Display( Dxyz );

303
    <An external ring residing in the CAS Maple>

304
    gap> Exyz := ExteriorRing( Qxyz, "e,f,g" );

305
    Q{e,f,g}

306
    gap> Display( Exyz );

307
    <An external ring residing in the CAS Maple>

308
  

309
  

310


311