Complete Subject Index

(0,1,all)-property 69
-geometry 436
0-diameter 82
1-diameter 82
2-cover, universal 77
2-design 71: - symmetric 56
2-dimensional Minkowski planes 1341
2-simple-connectedness 537
2-truncation 196
3-manifolds, hyperbolic 896
3-reflection lemma 958, 963: - affine 978; - projective 968
3-reflection theorem 958: - affine 978; - projective 968
3-regularity 413
4-gonal family 396
600-cell 871
absolute 78
absolute bound for strongly regular graphs 439
absolute space 663
addition 819: - of free angles 819
addition formula for Gegenbauer polynomials 852, 900
ADJACENCY 65, 483: - between chambers 484, 488, 1157; - i- 75, 484, 488; - - 1163; - in graphs 436; - of chambers 65, 678; - thin 1162
adjacency matrix 856, 879: - 439
adjoint of DLS 202
adjoint map 702
admissible 664
affine diagram 1172
affine elation 147, 173
affine geometry 6
affine homology 147, 173
affine incidence structures 744
affine line 1274
affine metric planes 980
affine perspectivity 147, 173
AFFINE PLANE 34, 70, 83, 744, 750, 1147, 1279: - completely projectively inflated 131; - derived 1329; - Desarguesian 467, 770, 971; - free 762; - free extension process 754; - nondegenerate 754; - obtained from projective plane 34; - over associative rings 1074; - Pappian 770, 971, 1311; - partial 744; - perspectivities 767; - projection pencils 767; - topological 1266; - -free constructions of 750; - -hyperfree elements in a partial 752; - -open 763
affine ring plane 1076: - coherent 1077
affine root 1095
affine section 1204, 1210
AFFINE SPACE 9, 33, 195, 470, 724, 796, 1147: - corresponding to vector space 971; - Desarguesian 971, 1156; - missed 197, 271; - nearfield 201; - Pappian 971, 989; - partial 794; - topological 1266; - underlying strong chain space 826; - weak 795
affine triple system 235
affinity 798, 1338
affino-projective space 234
Alexandrov neighbourhood 937
Alexandrov theorem 924, 925
algebra 788, 813: - associated with quadratic form 960; - inseparable 960; - kinematic 813; - local 788; - quadratic 813, 817; - quadratic - 960; - semilocal 834; - semiprimary 789, 797; - separable 960
algebraic group, split 634
algebraic groups, representation of 712
algebraic space 219: - full 219, 270
all modular hyperplanes theorem 212
alphabet 307
amalgam 617, 773: - classical 613; - noncollapsing 611
amalgamated sum 610
amalgamation property for partial models 774
ambient space 407, 408
ample connectedness 71, 656
André geometry 97
André plane 151
André system 151
angle 819, 974, 999: - free 819; - oriented 974, 980, 999; - Q-conform 974; - Q-relation 974
anisotropic kernel 635
anisotropic space 561
anisotropy 1100
annihilator 43, 1120
anti-automorphism 559: - of matrix ring 962; - of quadratic algebra 813
anti-Cauchy--Schwarz inequality 848, 896
anti-Cauchy--Schwarz property 891
anticollineation 835
anticorrelation 835
antiflag 435
antiflag-transitivity 67
apartment 21, 88, 479, 526, 1088, 1095, 1189, 1242: - diameter 91; - large 90
Apollonius problem 1350
approximation of unit sphere 874
arc plane 1298, 1299
Archimedean solid 604
Arguesian law 1122
arithmetic group, parabolic 1110
Artin ring 1039
Artin--Schreier characterization 1311
Artinian plane 932: - space 928
Artinian vector space 954
assignment 742
association of -DLS's 204
association scheme 359
asymptote 973
AUTOMORPHISM 32, 41, 77, 355, 771, 833, 837: - axial 355; - causal 935, 937; - central 248, 355, 1330; - continuous 1335; - discontinuous 1336; - F-semilinear 1261; - linearly induced 997; - of 2-design 354; - of abstract projective space 1261; - of affine metric plane 987; - of affine metric space 974, 996; - of affine plane 771; - of affine polar space 1195; - of affine ring plane 1078; - of Benz plane 248, 1329, 1335; - of building 1088; - of Burau geometry 837; - of chain geometry 791, 796, 806, 810; - of chamber system 1185; - of DLS 267; - of extension 757; - of free geometry 771; - of full matrix ring 1057; - of -gon 466; - of generalized Hughes plane 111; - of geometry 17, 77; - of icosahedron 878; - of integral lattice 911; - of Klein space 1002; - of Laguerre plane 771; - of lattice 1026, 1267; - of Minkowski plane 771; - of Minkowski product 828; - of module 784; - of Möbius geometry 824, 833; - of Möbius plane 771, 1337; - of projective metric space 990, 996; - of projective plane 771; - of projective space 32; - of quadratic circle geometry 992; - of space 650; - of spherical circular plane 1338; - of stable plane 1279; - of strong chain space 826; - of switching class 883; - of vector space 955
AUTOMORPHISM GROUP 826: - acting transitively on independent m-sets 275; - classical 1180; - exceptional 1181; - locally classical 1181; - locally exceptional 1181; - of compact (connected) projective plane 1288, 1291; - of DLS 268; - of Minkowski plane 1340; - of Möbius plane 1338; - of Laguerre plane 1346; - of the first kind 826; - positive 276; - topology of 1280
autotopism 148
autotopism group 148
axiom 5: - maximality 1257; - of joining 1336, 1339; - of linearity 544; - of parallel projection 1339; - of touching 1336; - richness 1336; - set of axioms 743
AXIOM SYSTEM: affine plane 34, 744; - affine polar space 724; - affine space 724; - apartment 88; - Bachmann place 1003; - Barbilian domain 1075; - Benz plane 746, 1328; - block space 13; - blocking set 321; - BN-pair, see -- Tits system; - building 90; - buildings, geometrical 606; - chain space 792, 820; - circular space 792; - Clarke 1022; - closure space 198; - connectedness relation 1021; - connection structure 1020; - contact space 793; - coordinatizable lattice 1125; - dimensional linear space 196; - Euclidean geometry 6; - Faulkner plane 1070; - generalized n-gon 387, 426, 520, 745; - generalized quadrangle 46, 387; - generalized regulus 807; - geometric lattice 1264; - Grzegorczyk 1021; - hyperplane of lattice 1124; - isoparametric map 1111; - Jordan homomorphism 831; - k-net 745; - Klingenberg space 1060; - Laguerre plane 240, 746; - Leiß ner plane 1079, 1080; - m-spread 144; - metasymplectic space 719; - metric product 1004; - Minkowski plane 240, 746; - Möbius plane 746; - Möbius space 793; - partial affine space 792; - physical geometry 1026; - point system 1124; - pointless metric space 1023; - polar space 45, 556; - principal denominator set 1054; - projective lattice geometry 1128, 1131; - projective plane 34, 744; - projective space 10, 39; - quadratic form 947; - quadratic set 815; - quasifield 141; - root system 699; - semilinear bijection 955; - stable n-space 1265; - Steiner system 748; - structured complex 526; - Tits system 563; - uniformity 1027; - V-plane 1051; - V-n-space 1045; - weak chain space 791; - weak Minkowski chain space 826; - Whitehead 1020
axis 968, 1049, 1282: - of inversion 249
axis C 249
Bachmann plane 1003
Baer cone 322: - of type 322
Baer group 179
Baer involution 169, 835
Baer p-element 177
Baer ring 1137
Baer subplane 115, 125, 158, 316, 835, 1286
Baer subspace 322, 452
Baer theorem 169
Banach spaces, reflexive 1264
Barbilian domain 1075, 1078, 1080
Barbilian plane 1036
Barbilian space 1036, 1045, 1135: - affine 1050; - dual affine 1050; - empty 1135
Bartolone--Ostrom plane 165
base of cone 318
base point 420
base-line 417
basic graph 485, 490
basis 200, 552, 784, 1097, 1129: - affine 976; - homogeneous 129, 1121, 1123; - max 73; - min 73; - of flat 1047; - of 835; - orthogonal 954
basis number, invariant 1040
Beckman--Quarles theorem 930
beetle figure 795
Benz plane 218, 229, 238, 250, 748, 767, 795, 821, 823, 1327: - classical 1331; - coherent 1332; - complete 237; - connected finite-dimensional 1333; - coordinatizable 242; - egglike 242; - embeddable 242, 1328; - finite-dimensional 1332; - free extension process 755; - locally classical 1334; - Miquelian 241, 250, 793, 802, 813, 821, 1328; - nondegenerate 756; - projective 748; - pseudo-pencil 767; - pseudo-perspectivity 767; - semiclassical 1334; - topological 1331; - valency of element 753; - -free construction for 751
Benz space 237, 254
Bernoulli numbers 1109
Bezout domain 1062
Bézout ring 1131: - proper right 1131
biassociativity 1263
bijection: -linear 955; - linear 955; - semilinear 955
bimodule 1136: - isomorphism 1139
biplane 1147
Birkhoff--Stone theorem 1019
block design 13: - balanced incomplete 351
Block lemma 252
block space 4, 13, 66, 67: - topological 4
blocking set 68, 126, 321: - irreducible 127
BLOCKS 66, 86, 351: - base 356; - in Laguerre plane 746; - in Minkowski plane 746; - in Möbius plane 746; - in Steiner system 748; - initial 356; - of design 351; - of locally classical geometry 1226; - of semibiplane 86; - repeated 351
blueprint 481, 625, 626, 629, 637: - realizable 627, 629
BN-pair 385, 479, 549, 563, 688, 1309, see also Tits system: - split 591
BNB condition 90
Boerner semifield 154
Bolyai--Lobachevsky geometry 891, 895
Boolean space 211, 262
Borel measure 851, 874
Borel subgroup 20, 79, 479, 563, 567, 712, 1093, 1100, 1101, 1153, 1184, 1185: - finite 1184
Borel--Serre compactification 1101
Bose--Mesner algebra 360
boundary 504
Brianchon theorem 970
broken grid 413
Bruck net 441
Bruck--Chowla--Ryser theorem 354
Bruck--Kleinfeld theorem 638
Bruhat cell 519
Bruhat decomposition 564, 587: - refined 587, 629
Bruhat order 517
Bruhat ordering 519
Bruhat poset 715
Bruhat--Tits building 612
Buekenhout theorem 967
Buekenhout--Mäurer theorem 969
Buekenhout--Metz Hermitian arcs 315
Buekenhout--Metz unitals 125
BUILDING 15, 21, 69, 70, 88, 90, 526, 585: - affine 1096; - at infinity 1173; - compact (connected) 1309; - conforming to blueprint 627; - exceptional 592; - of algebraic group 568; - of spherical type 528; - of spherical type, spherical 22, 88, 578; - topological 1306
bundle 245, 997
bundle condition 233, 245, 822, 1328
bundle space 234
bundle theorem 233, 245, 986, 999
bundle with three property 220
bundle-equivalency 1346
Burau geometry 837
calotte 965
canonical space 1023
carrier 245, 248, 413, 765: - of flock 333
Cartan apartment 93
Cartan decomposition 1099
Cartan involution 1101
Cartan subalgebra 18
Cartan subgroup 1109
category 3
Cauchy filters 1026: - infinitely close 1026
Cauchy sequence 1024
causality 935
Cayley algebra 479, 572, 636
Cayley numbers 638
Cayley--Dickson process 1262
Cayley--Menger distance matrix 849, 895, 914, 915
Cayley-distance 990
centre 786, 968, 972, 993, 1049, 1282: - of homology 425; - of projection pencil 765
central projection 9: - centre of 9
centralizer, connected 1088
chain 790, 806: - 2- 882; - affine 796; - affine n- 836; - in lattice 1121; - K- 789; - m- 835; - of shape 1027
chain domain 1061
chain geometry 790, 1328: - affine 796, 817; - direct products of 830; - Miquelian 822; - n- 835; - over quadratic algebras 813; - restricted affine 800; - trivial 831
chain group 218, 260
chain ring 1132: - left 1132; - right 1132
chain space 794, 821, 825: - associated to quadratic set 816; - connected 793; - Miquelian 821; - nontrivial 793; - strong 825; - weak 793
chamber 17, 65, 75, 96, 479, 483, 488, 678, 1098
chamber system 65, 96, 488, 489, 509, 678: - connected 489, 593; - locally classical 1185; - locally finite 1185; - pointed 593; - simple m-connectedness 509; - strongly connected 489; - with group action 494
chamber system correspondence 679
chambers opposite 580
Chang graphs 443
Chang--Los--Suszko theorem 743
character 1087
character of semi-oval 317
characteristic 948, 990, 1148, 1169
characteristic set 263: - algebraic 264
Chevalley basis 702
Chevalley group 466, 479, 702, 703, 710: - twisted 711
circle 83, 249, 836, 981: - belonging to bundle 997; - Euclidean 1337; - in Burau geometry 836; - in Laguerre plane 240; - in Minkowski plane 240; - oriented 1339; - proper 991; - r- 1002; - regular 981, 992, 994, 996; - singular 981, 994
circle plane 237, 1327, 1336
circles opposite 249: - symmetric 249; - touching 992
circuit 436, 520: - geodesic 88; - in graph 386; - optimal 88
circular extension 1202
circular plane 237, 824, 1336
circular space 208, 794: - locally affine 795; - sharply locally affine 253; - strongly locally affine 253; - weak locally affine 253, 272, 1194, 1195, 1205
clan 398
class 261: - hereditary 262; - of 3-transpositions 726; - of k-root subgroups 727; - of k-transvections 727; - subgeometry-closed 261
classical domination 1270, 1286, 1308
claw bound 439
Clebsch graph 440, 463, 856, 905
Clebsch quartic surface 440
Clifford algebra 51, 698, 958, 1308
Clifford theorem 983
clique 651
closed set 416
closure 198, 416: - circle geometric 997; - lemma 650
closure space 73, 195, 198: - graded 73; - induced by restriction 199; - simple 198; - simplified 198
co-amalgam 773
coaffine plane 1275
cocentre 173
cocharacter 1087
coclique 275
cocycle, 2- 882
code 128, 307, 361: - GDRS 309; - GRS 309; - of affine geometry 365; - of projective geometry 365; - self-dual 362; - spherical A- 854; - tight 854
codeword 128
codimension 202, 483, 950
coding theory 13
cohomological dimension 1106
cohomology 882
coline 202
collar 669
collinear 414
collinearity 68, 435, 650
collinearity graph 68, 437, 651
COLLINEATION 41, 43, 113: - affine 1048, 1078; - axis 1282; - central 35, 1282, 1330; - centro-axial 1282; - continuous 1335; - fixing Von Staudt chain 837; - h-affine 1050; - in Leiß ner plane 1079; - of affine metric plane 987; - of affine metric space 974, 976; - of affine plane 43; - of Burau geometry 837; - of chain geometry 810; - of compact (connected) projective plane 1288; - of derived affine plane 1330; - of derived projective plane 1330; - of Figueroa plane 113; - of generalized quadrangle 420, 424; - of L-plane 1079; - of projective space 1136; - of quadric 815; - of stable plane 1279; - of translation plane 172; - of Veldkamp space 1048; - planar 177; - semilinear induced 965
COLLINEATION GROUP 42, 145, 183: - planar 177; - semitransitive relative to 183; - tangentially transitive relative to 183; - transitive relative to 183
comet 1173
commutator map 631
commuting tori graph 729
compactification 1105, 1263: - theorem 1100
compactness in lattices 1120: - local 1262
comparison of parabolics 1183
complement 1120: - of graph 436
complete set of invariants 898
COMPLEX 65, 96, 483, 504, 521, 539: - admitting parameters 539; - belonging to M 521; - locally finite 539; - of Coxeter type 522; - over I 484; - pure 483; - realization of 96; - simplicial 482; - strongly connected 484; - thin 504; - thin-with-boundary 504
components: of spread 143; - of translation plane 143
composition of paths 599
condition 1039
cone 318, 847, 956, 961, 965: - elliptic 1328; - future- 936; - null 926; - of norm 0 847; - of singularity 797; - open 1269; - quadratic 237, 814
conference matrix 885
configurational postulate 823
configuration: induced 389; - tactical 820
conformal transformation 929
congruence 143: - B- 1080
congruence order 879
congruence relation 972, 988, 989
congruency 819
conic 57, 121, 802, 816, 1333: - irreducible 298; - on quadric 822
conjugations 1262
connected 436
connectedness 76, 1024, 1262: - relation 1021; - residual 76
connection relation 1020
connection structure 1020
connectivity, finite 1279
contact relation 795: - geometric 811; - natural 795, 799, 811
contact space 795
containment relation 1128
continuation along geodesics 587
continuity: of intersection 1265; - of join 1265
contractibility, local 1272
contraction 1272: - of closure space 203
convex hull 589
convexity 507: - condition 89; - space 97
coordinatizability 232: - F-unique 264, 266; - of lattice 215
coordinatization 37, 256: - F- 256; - of DLS 256; - of lattice 1125; - of projective plane 35
coplane 202
copoint 202
copolar space 458
corank 75: - of subset 75
corner associated to parabolic subgroup 1103
corner condition 823
correlation 43, 77, 78, 359: - of designs 359
coset geometry 17, 20, 79
coset, left 17
cotriangle space 457, 458
cotype 75, 484
countability: first 1262; - second 1262
cover 77: - 2- 77; - proper 77; - universal 2- 1161, 1178, 1187; - universal m- 77
COVERING 324, 1171: - between abstractive sets 1020; - minimal 324; - of Barbilian planes 1071; - of Barbilian planes, universal 1071; - of closed set by closed set 199; - of PSL 1301; - of PSU 1312; - of SO 1296; - reduced, see -- minimal
Coxeter complex 1171
Coxeter diagram 83, 500, 1160: - spherical 1160
Coxeter geometry 22
Coxeter graph 905, 912
Coxeter group 502, 517, 563, 679
Coxeter matrix 500
Coxeter part of diagram 1148
Coxeter stroke 1148
Coxeter system 502
Coxeter--Killing transformation 870
Coxeter--Tits complex 480, 500, 518, 624, see also Coxeter complex: - belonging to M 503
Cremona transformation 804
Cremonian geometry 804
cross of lines 965
cross ratio 59, 787, 818
cross-polytope 867, 904: - truncated 904
cube 877: - twisted 256
cubic: lattice 865; - twisted 298
cuboctahedron 866
curve 829: - Hermitian 123; - rational 801
curve of degree r: rational normal 801
curve space 829
cycles 1339
cylinder: oval 242; - semi-oval 242
cylinder plane 1268, 1274, 1295, 1299, 1300: - dual 1299; - modified real 1296; - modified real dual 1276; - skew 1295
Danzer's problem 85
deck transformations 594
decomposition of motions, canonical 979
deep holes 912
deficiency 328, 1149: - 0- 1149; - 1- 1149
degree 438: - of homogeneity 1281, 1301; - of representation 812
Dembowski semi--spaces 254
Dembowski semiplane 254
Dembowski--Wagner theorem 210
Dempwolff group 179
Dempwolff plane 161, 162
denominator: planar 1054
denominator set: planar, see -- principal; - principal 1054
dependence 849
derivable plane 157
derivation 156, 168, 338, 702: - inner 883
derived plane 157: - topological 1332
derived quasifield 157
Desargues axiom 11, 69
Desargues condition 1079
Desargues configuration 91, 213
Desargues postulate 1122, 1130
Desargues theorem 31, 37, 109: - dual 220
Desarguesian plane 969: - locally 1300
Desarguesian projective space 236
Desarguesian space 270
DESIGN 13, 351, 352, 360, 387: - affine 2- 56; - affine resolvable 2- 56; - antipodal spherical 867; - block schematic 360; - derived 352; - divisible 372; - Euclidean t- 875; - Hadamard 3- 56; - matroid 195; - quasi-symmetric 360; - regular geometric 195; - residual 352; - resolvable 352; - spherical t- 875; - square 352; - symmetric 352; - symmetric 2- 56
De Sitter space 848
De Sitter space-time 930, 937
diagonal axiom 451
diagonal clique 451
diagonal points property 220
diagonal subtorus 1096
DIAGRAM 48, 81: - basic 81; - hyperbolic 1171; - linear 85; - maximal 1148; - minimal circuit 1148; - of almost finite type 1157; - of Coxeter type 1149; - of finite type 1157; - of infinite type 1157; - of singular type 1149; - of strongly finite type 1161; - of strongly infinite type 1161; - special 82, 1148
diameter 386, 437, 520, 688, 1023, 1157: - of building 528; - of shadow space 688
Dickson nearfield 151
Dickson--Knuth semifield 155
difference families 356
difference set 355
digon 48
dilatation 250, 786, 798, 979, 1049: - affine 1050; - dual a-affine 1050
Dilworth completion 217
Dilworth truncation 217
DIMENSION 196: - affine 971; - of affine space 971; - of affino-projective space 234; - of chain space 795, 797; - of closed set 416; - of flat 1047; - of lattice of varieties 200; - of poset 485; - of projective lattice geometry 1129; - of projective space 552; - of representation 812; - of translation plane 146; - of variety 196; - projective 964, 1259; - small inductive 1276
dimension formula 1281
dimension function 1121
dimension inequality 1264
dimension lemma 211
dimensional linear space, see DLS
direct product 652: - of chamber systems 1185
direct sum 80: - of DLS's 204; - of geometries 38
direct sum theorem 81, 1160
direction 798: - of Q-symmetry 976
disconnectedness, total 1262
discreteness 1262
discriminant 911
disjoint relation in lattices 1120
distance 923, 1023, 1027: - elliptic 875; - hyperbolic 891; - in Barbilian space 1135; - in physical space 1027; - Q- 972
distance function 807
distance matrix 913: - hypermetric 915; - of negative type 913
distant relation 1043, 1076, 1079, 1135
distributor 147
division ring 8, 30, 559: - locally compact 1263; - topological 1260
DLS 196, 198, 206, 208, 216, 223, 260, 263, 1191, 1265: - algebraic 263; - basis-transitive 273, 275; - Boolean 208, 224; - circular 208; - completely trivial 208; - connected 206; - coordinatizable 263; - design 209, 223; - F-algebraic 263; - finitary 200; - flag-transitive 271; - highly transitive 267; - hypercircular 223; - hypercircular t- 208; - hyperplane-antiflag-transitive 273; - -fold circular n- 222; - irreducible 206; - Jordan-transitive 270; - k-connected 206; - m-closed 216; - n- 196, 257; - n-homogeneous 268; - n-modularity complemented 257; - noncircular 208; - nonregular 223; - nonregular hypercircular 209; - point-antiflag-transitive 273; - point-primitive 275; - regular 223, 260; - regular n- 221; - regular hypercircular n- 209; - sharply locally projective 211; - strongly locally affine n- 265; - strongly locally projective 211; - strongly locally projective n- 265; - supersolvable n- 257; - thick 208; - top-thick 266; - trivial 223; - trivial n- 208; - type x-transitive 268; - unimodular 260; - voltage-graphic 262
dodecahedron 869, 890
domination by classical planes, see classical domination
dot product 923
double coset 564
double space 99
Dowling lattice 212, 213
Dowling space 262
duad 401
DUAL 72: - affine Barbilian space 1050; - algebraic hypersurface 303; - basis 1041; - basis of flat 1047; - code 308; - cylinder plane, modified real 1296; - degree 371; - design 448; - distance 371; - eigenmatrix 360; - Euclidean plane 1295; - grid 393; - Hermitian plane 1295; - lattice 911; - linear space 465; - Minkowski plane 1295; - net 465; - of classical generalized quadrangle 633; - of complex 636; - of design 351; - of DLS 203; - of function space 218; - of generalized polygon 387; - of hyperplane space 72; - of lattice of n-DLS 202; - of modular lattice 202; - of projective space 202; - of projective space over ring 1042; - of topological projective plane 1274; - pairing 1036; - partial quadrangle 333; - polar graph 676; - polar space 676; - statement 1045
duality 43, 78, 117: - for k-arcs 305; - for projective spaces 1044; - H- 118; - orthogonal 202
Dynkin diagram 19, 571, 700, 905: - extended 905; - relative 550, 1101
eigenvalue 356, 439, 442
Einstein's cylinder universe 237, 929, 937
elation 35, 42, 376, 420
elation generalized quadrangle 420
elation group 420, 1347
element 785: - Euclidean 952; - geometrical 1021; - n-generated 1124, 1129; - singular 952; - unimodular 785; - zero 849; - -irreducible 1019
elementary divisor 363
elementary divisor theorem 1098
elementary group 1038
elementary-homotopic paths 599
elementary homotopy 512: - of labels 627; - of sequences 628
elementary Weyl homotopy 525
ellipse 802
ellipsoid 886
elliptic plane 875, 1294
elliptic quadric 309, 451, 455, 469
elliptic space 847, 879, 949
embeddability 125: - F-unique 264; - universal 235
embedding 78, 255, 436, 654, 743, 773: - C- 265; - canonical 789; - F-rigid 266; - F-unique 266; - F-universal 266; - flat 436; - full 436; - generalized projective 256; - joint 773; - lattice 255; - of chain geometry 831; - of geometry 78; - of linear space 232; - of negative definite space 914; - of planar space 232; - projective 256; - strong 267, 1268
embedding theorem 951, 1268
enough modular hyperplanes theorem 212
entourages 1027: - fundamental system of 1027
epimorphism 77, 425: - of generalized 3-gon 425
equiangular lines 858
equidistant sets of elliptic points 881
equimetry 955, 961, 975, 991
equivalence: affine 799; - thin 1162
erection 202
Erlangen Program 16, 845
Euclid's axiom 845
Euclid's postulate 794
Euclidean plane 295, 819, 964, 1003: - dual 1295, 1299
Euclidean projections 911
Euclidean space 849, 948, 1029, 1327
Euler characteristic 1092
Euler graph 880: - labelled 881
Euler--Poincaré characteristic 1108
Euler--Poincaré measure 1108, 1109
eumorphism 1053
eutactic star 864, 889, 890
exceptional 3-symmetric plane 1302, 1306
exchange axiom 199
exchange condition 503, 536
extension 753, 1145: - étale 1099; - free 759; - generated over model 753; - i-fold c- 209; - i-fold L- 209; - n-times linear 1224; - of design 352; - totally ramified 1263; - -free generated 754
extension process 772: - faithful 772
extension series 753: - free 759; - -hyperfree 754
extension theorem 619, 637, 959: - for twin buildings 637
exterior algebra 52
exterior hyperbolic space, see De Sitter space
exterior product 808
exterior set 338
F-universality 267
face 96, 483: - associated to parabolic subgroup 1103; - of complex 96
factorization of groups 1153
failed bundle DLS 258
failed Desargues space 258
failed Pappus space 258, 263
fan 245
Fano plane 197: - free 765
Faulkner plane 1036, 1070
FDLS, supersolvable 217
Feit--Higman theorem 539, 1149, 1307
fibre bundle 1346
field 55: - alternative 631; - Galois 55; - mereological 1021; - of left quotients 1125; - ternary 1271
fifteen schoolgirls problem 327
Figueroa planes 112
filter, completely prime 1019
filtration 1092
finite character property 73
firmness 491
first main characterization of buildings 532, 548, 595
Fischer group 726
Fischer plane 166, 167, 235
Fischer space 229, 235, 236, 677, 727: - generalized 728
Fisher inequality 353
flag 17, 47, 65, 75, 271: - opposite 418; - reduction of 94
flag complex 96, 485
flag graph 437
flag space 1293
flag-diameter 437
flag-transitive linear space 469
flag-transitivity 17, 66, 67
flat 30, 277, 913: - bases of 1047; - distant 1048; - dual 1047; - dually distant 1048; - empty 1047; - k- 1264; - of affine space 33; - spanned by independent points 1047; - transversal 1047
flexibility 1338
flexible plane 1290
flock 72, 166, 248, 333, 397: - linear 167, 333
flocks 336, 337, 338, 340, 1329: - derived from generalized quadrangles 398; - exceptional 335; - of Bader, Lunardon and Thas 337; - of cones 336; - of Fisher 337; - of Fisher, Thas and Walker 336; - of Ganley 337; - of hyperbolic quadrics 334; - of Johnson and Payne 337; - of Kantor 336, 337; - of ovoids 334; - of Payne 337, 338; - of Payne, Thas, Johnson, Lunardon and Wilke 337; - of Thas 335; - sporadic 340
folding 506, 1200, 1226: - belonging to panel 508; - opposite 578
forbidden subgeometry theorems 259
FORM: 2-singular 952; - alternating 665, 886; - anti-Hermitian 665; - Artinian 954, 961; - bilinear 44, 803, 948; -- alternating 44; -- symmetric 927; -- symmetric self-dual 362; -- symmetric self-orthogonal 362; - elliptic 948, 952; - Euclidean 948, 952, 953, 990; - Galilean 952; - Hermitian 559, 635, 665, 670, 728; -- alternating 560; -- trace valued 560; - hyperbolic 948, 952; - irregular 950; - isotropic 950; - K-linear 813; - linear 949; - Minkowskian 948, 952, 961, 965, 990; - nondegenerate 664; - orthogonal 886; - pseudoquadratic 635, 670; - quadratic 45, 52, 363, 560, 670, 690, 813, 947, 948, 1197; -- Hermitian 522; -- nonsingular 45; -- split 52; - reflexive 664; - regular 950; - sesquilinear 44, 1308; -- Hermitian 44; - singular 950, 952; - symmetric 665, 1294; - unitary 886; - -anti-Hermitian 665; - -Hermitian 665; - -quadratic 670; - -sesquilinear 664; -- associated to polarity 665; -- proportional 665; -- reflexive 559; -- trace-valued 666; - -Hermitian 665; - -quadratic 670
forms: equimorphic 955; - geometrically equivalent 955; - isometric 955; - isomorphic 955; - proportional 671
formula 742: - atomic 742; - universal 743
Foulser planes 160
foundation 481, 625: - supporting building 626
frame 1018, 1052: - H- 714; - m- 835; - orthogonal 864; - polar 557; - spatial 1019
free extension 742
free set 73, 74
Frobenius automorphism 706
Frobenius--Burnside lemma 354
fully projective set 657
function space 218
function, -invariant 868
functional space 97
functor 3
functoriality 1088
fundamental group 600
fundamental representation of G 713
fundamental theorem: of affine geometry 925; - of affine metric geometry first 974; - of affine metric geometry second 976; - of projective geometry 562, 965; - of projective metric geometry first 990; - of projective metric geometry second 991; - of quadric circle geometry 992; - on stereographic projection 995
-gon 82
-gon 437: - regular 464
-pair: distance transitive 466; - line distance transitive 466; - line geodesic transitive 466; - locally i-arc transitive 466; - point distance transitive 466; - point geodesic transitive 466; - weakly distance transitive 466; - weakly geodesic transitive 466
gallery 483, 489: - closed 483; - connected 483; - simple 483
Galois automorphism 690
Galois cohomology 1101
Galois extension 1089, 1099
Galois field 933
Galois geometry 57
Galois group 571, 1089, 1099
gamma space 651, 654
gamma space lemma 651
Ganley semifield 155
gate property 537, 576, 589
Gauss--Bonnet theorem 1108
Gegenbauer coefficients 857, 858, 859, 860
Gegenbauer polynomial 853, 862, 900, 901, 902
generalized -gon 521
generalized 2m-gon, pseudo-classical 1226
generalized André plane 149
generalized André system 149
generalized digon 81, 387, 467, 485, 521
generalized g-gon 82
generalized Hall plane 158
generalized Hall system 158
generalized hexagon 54, 387, 460, 661, 689, 1230: - classical 391; - Moufang 424
generalized homology 420
generalized Hughes plane 112, 115
generalized m-gon 21, 520, 585
generalized -gon 625
generalized n-gon 426, 745, 750, 768: - free 385; - free extension process 754; - groups of projectivities 768; - nondegenerate 755; - order 387; - perspectivities 768; - projection pencils 768; - -hyperfree chain in a partial 752
generalized octagon 387, 661, 709: - classical 392; - Moufang 425
generalized plane, -free construction of 750
generalized polygon 387, 464, 466, 520: - Moufang 426; - topological 1306
generalized quadrangle 46, 334, 369, 387, 441, 453, 460, 465, 467, 660, 768, 1196, 1230, 1240, 1339, 1343, 1349: - 3n-dimensional 1350; - antiregular 1349; - classical 389, 448; - derivation of 1349; - Desarguesian 417; - elation 1307, 1310; - embedded in affine space 408; - embedded in projective space 407; - Moufang 423; - -transitive 424; - perspectivities 768; - projection pencils 768; - projective 407; - Roman 399; - topological 1349; - weak projective 408
generalized triangle 387, 465, 521
geodesic 507, 628, 1090: - based at point 437; - maximal 437
geodesic action 1102
geodesic condition 88
geodesic transitivity 1150
geodesically closed 652
geometric group 271, 276
geometric lattice 12, 1147
geometric realization 1089
geometric t-spread 325
geometrical realization 498, 555
geometrically irreducible subgroup 171
geometrically primitive subgroup 172
geometries, homogeneous classes of 777
GEOMETRY 47, 75, 76, 222, 1166, 1226: - affine-dual-affine 1215, 1218, 1223; - affine metric 970, 971; - Alt(7) 89, 523, 687, 1166, 1178, 1203, 1211; - at infinity 992; - - 240; - - 829; - belonging to diagram 48, 81; - c L- 222; - chain, see chain geometry; - circle 470, 988; - classical projective 1129; - combinatorial 195, 201; - continuous 1121; - coset, see coset geometry; - cycle 254; - d-injection 241; - Desarguesian 1130; - dual 82; - elliptic 845; - Euclidean 6, 845; - F-squashed 278; - finite 13; - finite projective 13; - finitely generated 1129; - firm 47, 76; - flag- 1226; - flat 49; - flat anomalous 1167; - hyperbolic 845; - inner 991; - - 764; - locally classical 1166, 1226; - locally finite 1157; - locally pseudo-classical 1226; - minor 1129; - module-induced 1129; - (n-)cluster 1129; - noncommutative 764; - non-Euclidean 845; - nonparabolic 728; - of Coxeter type 547; - of grade n 218; - ordered space 1269; - partial 1244; - permutation 271, 276; - permutation of rank s+1 278; - physical 1025; - plane Euclidean 980; - point-irreducible 1134; - point-uniform 1132; - pointless 1017; - projective 1117, 1208; - projective (lattice) 1119, 1128; - projective metric 990; - pseudo-affine 1208; - pure 81; - quadric circle 992; - quotient 1129; - reduced 1129; - reflection 988; - spherical 845; - squashed 241, 271; - squashed of rank n 278; - stable 1135; - thick 47, 76, 222; - thin 76; - tilde 1239; - topological 7; - transversal 47; - with group universality 775; - with homomorphism 1060; - without points 1017; - Z- 1130
Gewirtz graph 313, 333, 441, 463
girth 437, 1158: - geometric 437
Gödel's completeness theorem 743
Golay code 854, 859: - extended binary 361; - extended ternary 372
gonality 82, 437
Gosset polytope 500, 855, 871
Gram matrix 846, 864, 866, 878, 894, 898, 902, 910: - doubly stochastic 895
GRAPH 386, 436, 651, 721, 855, 1159: - associated to model 749; - bipartite 386; - complete 436; - distance regular 438; - generalized line 904; - geometric 442; - half dual polar 678; - locally Petersen 1231; - locally 721; - pseudo-geometric 442; - reduced 652; - reflexive 907; - regular 438, 856, 1158; - regular two- 878, 885, 905; - simple 216; - strongly regular 438, 856, 1159; - two- 881
graphs, switching equivalent 880
Grassmann coordinates 392, 1261
Grassmann geometry 1204: - affine 1208; - affine d- 1201
Grassmann line 49
Grassmann manifold 1261
Grassmann map 809, 810, 814
Grassmannian 46, 49, 653, 808, 836, 1260, 1348
Grassmannian characterisation 718
grid 393, 721
GROUP 16, 374: - affine 374; - almost simple 374, 469, 667; - anti-unitary 1295; - classical 18, 667; - extraspecial 171; - of Lie--Chevalley type 22; - of type (FL) 1106; - planes 1298; - projective 18; - quasiperspective 1298; - simple 18; - symplectic 18; - topological transformation 1280; - unitary 1294
group schemes 1100
group universality 775: - for geometries 775; - theorem 773
H-configuration 118
Hadamard 3-design 376
Hadamard design 275, 369, 467
Hadamard matrix 352, 860
Haemers partial geometry 446
Haemers--Roos inequality 388
hair 1173
half Moufang property 468
half ordered plane 100
half way increase 224
half-plane 1279
half-spinor geometry 53
half-spinor space 53
Hall plane 125, 158
Hall system 158
Hall triple system 230, 235, 1192
Hamada's conjecture 368
Hamming distance 307, 361
Hamming graph 461
harmonic polynomial 850
Hartmanis partition lattice 224
Hasse diagram 715
Hasse--Weil theorem 58
Hausdorff space 1266, 1276
height 199
Helmholz--Lie problem 12, 1269
hemisystem 332
Hering plane 161, 165, 374, 470, 1156
Hering space 374, 470, 1156
Hering types of Möbius planes 1337, 1338
Hering--Ott--Schaeffer plane 161
Hermite 2-graph 886
Hermitian arc 314, 316, 317, 323
Hermitian curve, nonsingular 315
Hermitian matrix 1066
Hermitian plane 1294, 1302, 1305, 1306: - connected complex 1302; - modified 1295, 1302; - quaternion 1297
Hermitian space 1268: - nondegenerate 561
Hermitian unital 275, 352, 374, 470, 1155
Hermitian variety 369, 390, 456, 1169, 1198, 1199: - nonsingular 320
Heron's theorem 210
hexagon 905: - inscribed 969; - Pascalian 969
Higman inequality 388, 442
Higman--Sims family 440
Higman--Sims graph 440
Hilbert space 1264
Hilbert--Smith conjecture 1282
Hjelmslev diagram 99
Hjelmslev geometry 1063, 1132, 1133: - point-Noetherian weak 1133; - weak 1132
Hjelmslev plane 100, 933, 1311: - locally compact connected 1311
Hjelmslev ring 788, 1064, 1133: - Artinian 1133; - left-weak 1132; - Noetherian 1133
Hjelmslev--Klingenberg geometry 99, 1063
Hjelmslev--Moufang plane 1065
Hoffman--Singleton graph 440, 463, 466, 1149, 1150, 1154, 1238
homeomorphism: of circle 1336; - of topological projective spaces 1262; - of vector spaces 1262
homogeneity 776: - flag- 1293; - line- 1294, 1298; - point- 1268, 1294, 1298; - theorem 776
homology 35, 43: - generalized 425
homomorphism 77: - full 1053
homothety 250
homotopy 512, 599: - of stable planes 1278
homotopy group 593
homotopy type of building 1091
host space 256
Hughes plane 110, 1292, 1303
Hughes--Kleinfeld semifield 155, 159
Hughes--Kleinfeld semifield planes, dual 160
hull: affine 971; - convex 574
hyperbola 802
hyperbola structure 1330, 1339
hyperbolic plane 891, 1258, 1279, 1300: - exterior 1294; - interior 1294, 1299; - modified real 1296; - united 1295
hyperbolic quadric 443, 451
hyperbolic space 848, 949, 1268
hypercube 555, 889: - halved 889
hypergraph 66, 351
hyperline 606
hyperoctahedron 889
hyperoval 120, 298
hyperovals of Cherowitzo 300
HYPERPLANE 30, 71, 200, 212, 323, 654, 816, 836, 949, 950, 1124: - affine 70; - at infinity 972; - dually distant 1045; - geometric 70; - in Barbilian space 1045, 1135; - in geometric lattice 1264; - in Klingenberg space 1060; - in projective space 964; - modular 212; - of modular lattice 1124; - of projective space over ring 1042; - osculating 298, 811; - projective 70; - regular 1124; - tangent 316, 317
hyperplane space 71
i-order 83
i-secant 318
-shadow 604
icosahedron 857, 866, 867, 869, 871, 877, 883
icosidodecahedron 869
ideal 698
identities 764
INCIDENCE 6, 109, 112, 124: - deduced from t-spread 325; - in affine plane 744; - in Barbilian space 1045, 1135; - in DLS 198; - in Figueroa plane 112; - in general hexagon 391; - in generalized n-gon 745; - in k-net 745; - in projective plane 109, 744; - in projective space over ring 1042; - in shadow space 686; - in Steiner system 748; - in topological projective plane 1284
incidence complex 85
incidence geometry 3: - metric 3; - ordered 3; - topological 3
incidence graph 68, 386, 437, 749
incidence group 100: - topological sliced 1269
incidence matrix 352
incidence number 436
incidence relation 47, 75
incidence structure, internal 1329
incidence system 684
inclusion 1018, 1020
independence space 195, 201
independence with finite character 200
independent family 1121
independent sets 74
index 568
infinity: hyperplane at 33; - line at 9; - plane at 9; - point at 9; - subspace at 9
inherited group 157
inner distribution 360
inner product 851: - indefinite 891
inner product space 846
inscribed angles theorem 981
instant 1018
interior construction 749: - restricted 749
interlacing 357
internal plane 310
interpretation function 742
intersection 1259
intersection array 438
intersection property 93, 95, 682: - for flags 605; - for objects 605
intersection set 419
interval causal 937
interval of x over y 1120
invariant 16: - basic 869; - factor 363
inverse path 599
inversion 249, 800, 824, 895, 996, 1337: - B- 249; - elliptic 996; - hyperbolic 996
inversion principle 228
inversive plane, see Möbius plane
involution 787: - Baer 1282; - free 1282
irreducible chain 745: - closed 745
isometry 928, 955, 961, 975: - in elliptic space 877
ISOMORPHISM 33, 77, 832, 833: - between angle group and 819; - between Sp(4,3) and U(4,2) 1197; - between spaces of polynomials 851; - between Sym(6) and Sp(4,2) 1196; - fundamental 832; - induced by K-Jordan isomorphism 833; - of affine chain spaces 832; - of affine metric spaces 974, 976; - of affine ring planes 1078; - of affine spaces 33; - of bimodules 1138; - of chain geometries 806; - of chain spaces 817; - of closure spaces 199; - of collinearity graphs 663; - of (combinatorial) simplicial complexes 555; - of generalized quadrangles 400; - of geometries 77; - of lattices 1019, 1124, 1126, 1127, 1140; - of metric vector spaces 962; - of models 776; - of modules 1139; - of ppm-spaces 1025; - of projective metric spaces 990, 995; - of quadratic algebras 960; - of quadratic circle geometries 962, 992; - of spaces 650; - of spherical buildings 619; - of stable planes 1279; - of structured complexes 527; - of submodule lattices 1119; - of two affine planes 147; - of vector spaces 955
isoparametric foliation 1112
isoparametric map 1111
isoparametric subvariety 1112
isosceles triangle theorem 986
isotopism 147
isotropic vector 561
isotropy group 1112
Iwahori subgroup 1098, 1101
Iwasawa decomposition 1099
J-residue 489
J-shadow 682
J-star 489, 1185
Jacobi identity 698
Jacobi vector field 1111
Jacobson radical 789, 813, 1038, 1122
Johnson plane 159
Johnson scheme 360
Johnson--Walker plane 159
join 483, 652, 1120, 1259
join geometry 97
join-irreducibility 1121
Jordan algebra 1066
Jordan automorphism, K- 832
Jordan curve 1337, 1341
Jordan decomposition 1092
Jordan group 270
Jordan homomorphism, K- 832
Jordan isomorphism, K- 832
Jordan pair 1067
Jordan product 1066
Jordan set 270: - improper 270
Jordan theorem 1279
Jordan--Dedekind chain condition 201
Jordan--Dedekind condition 73
Jordan--Schoenflies theorem 1337
Jordan-transitivity 270
k-arc 119, 298, 301: - complete 298; - embedding 301
-arc 119, 120: - complete 120; - maximal 120; - trivial 120
k-cap 309: - complete 309
k-group 567
k-net 745, 767: - free extension process 755; - nondegenerate 755; - partial 746; - satisfying Thomsen condition 770; - -free construction 751; - -hyperfree elements in a partial 753
-Steiner system, -hyperfree elements of a partial 753
k-parabolic 568
Kac--Moody group 591, 618: - split 592
Kalscheuer nearfield 1290, 1291, 1303
Kalscheuer nearfield plane 1305
Kantor embedding theorems 266
Kantor inflation trick 1157
Kantor plane 164: - flag-transitive 164; - of type I 164; - of type II 165
Kantor semifield 154
Kantor theorem 211
Kantor uniqueness lemma 1268
kern 142
kernel 142, 298, 421, 1026, 1329, 1340, 1344: - anisotropic 568; - of normal rational curve 304; - of quadratic form 951; - of quasifield 142; - of region 1026; - of translation plane 142, 146
Killing form 1110
kinematic space 817
Klein correspondence 327, 403
Klein model 892, 895
Klein plane, real 1003
Klein quadric 51, 53, 558, 810
Klein space 1000
Klingenberg plane 1060
Klingenberg space 1058, 1060
Knarr generalized quadrangle 399
knot, see nucleus
Knuth semifield 154, 155
Kostant's conjecture 729
Krein conditions 439, 442
Krein inequalities, see Krein conditions
Kurosh--Ore dimension 1121
labelling 586, 626, 627: - consistent 627
Laguerre algebra 788, 793, 813, 826
Laguerre geometry 792, 826
Laguerre i-structures 254: - special 254
Laguerre plane 195, 237, 238, 239, 240, 251, 252, 746, 767, 795, 802, 814, 1307, 1327, 1328: - 2-dimensional 1344, 1350; - 4-dimensional 1346, 1347; - classical complex 1347; - complete 238; - constructed from Möbius plane 1338; - elation 1310, 1350; - embeddable 251; - Miquelian 252, 1333, 1345; - nonembeddable 1345; - ovoidal 1328, 1344; - perspectivities of 767; - semiclassical 1335, 1344; - topological 1344; - with 5-dimensional automorphism group 1345
Laguerre space 195, 826
language, first-order 742
Laplace equation 850
Laplace operator 850, 899
Laplacian 1111
large set of designs 352
large set of t-designs 373
Latin square 369, 372
LATTICE 31, 34, 39, 199, 201, 1098, 1117, 1259: - algebraic 1120; - Arguesian 1120, 1122, 1130; - atomic 31, 39, 199, 552; - atomic semimodular 195; - atomistic 1121, 1264; - Boolean 208; - bounded 1123; - complemented 201, 1120; - complete 199; - cubic 86; - cyclic 1122; - Euclidean 910; - even 861, 916; - geometric 195, 199, 201, 1264; - graded 201; - indefinite 910; - integral 861, 908; - integral unimodular 909; - irreducible 1121; - locally modular 34; - matroid 850; - modular 31, 39, 552, 1117, 1120; -- with point-system 1118; - module-induced 1124; - of finite length 1121; - of geometric lattice 1264; - of order n 1121; - of type I 908; - of type II 908; - ordered 31; - partition 213, 224; - primary 1122; - ranked 39; - reflexive 911; - relatively irreducible 1121; - root 700, 911; - semimodular 1264; - semiprimary 1122; - topological 1259; - unimodular 861, 908; -- integral 911; - upper continuous 1120; - upper semimodular 199; - weakly distributive 1026; - weight 700
Laurent series 1263, 1333
law alternative 1065
Leech lattice 854, 861, 871, 890, 903, 910, 912
Leiß @ ner plane 1036: - affine 1079; - c-transitive 1079; - -transitive 1079
length of vector 949
Lenz axiom 794
Lenz--Barlotti class 139
Lenz--Barlotti classification 1289
Levi complement 1301
Levi subgroup 706, 713, 1090, 1102
Lie algebra 519, 698, 699, 709, 1087, 1269, 1281: - simple 699, 700; - special 709
Lie bracket 1111
Lie geometry 1307, 1347: - associated to Laguerre plane 1339, 1349
Lie group 13, 18, 1281, 1301: - exceptional 479; - linear 18; - orthogonal 18; - quasisimple 1301; - semisimple 1342; - simple 18; - simple complex 18
Lie topology 1303
Lie transvection F-plane 1073: - coordinate ring of 1073
Lie triple system 1304
Lie--Chevalley group 15
Lie--Johnsen group 1312
lifting 78, 509
light cone 913
likeable mapping 162
likeable plane 162, 398
LINE 10, 94, 109, 111, 124, 325, 606, 651, 816: - absolute 118; - affine 724; - as manifold 1277; - at infinity 141; - coaffine 1275; - coherent 1077; - concurrent 387, 435; - cross 1076, 1079; - elliptic 875; - equiangular 855, 902; - Euclidean 959, 1337; - extended 997; - external to k-cap 309; - fixed 391; - hyperbolic 58, 395, 891; - improper 141; - in affine ring plane 1076; - in Alt(7)-geometry 1166; - in Barbilian space 1135; - in block space 210; - in design 56, 375; - in DLS 196; - in generalized n-gon 745; - in geometric lattice 1264; - in group coset geometry 395; - in Laguerre plane 240; - in Leiß @ ner plane 1079; - in Minkowski plane 240; - in projective plane 109; -- over octonion algebra 1067; - in projective space 551, 964; - in stable plane 1270; - in vector space 950; - interior 111; - isotropic 1294, 1297; - K- 1000; - l-distance 810; - null 925; - of affine plane 744; - of affine space 971; - of block space 68; - of Desarguesian topological projective 1263; - of embedded shadow space 715; - of Euclidean space 849; - of generalized Hughes plane 111; - of geometry quadrangle 521; - of inner product space 847; - of 318; - of k-net 745; - of Klein space 1000; - of Lie geometry 1349; - of locally classical geometry 1226; - of perspectivity 417; - of projective plane 744; - of space 649; - of Sperner plane 1004; - of translation plane 141; - on Barbilian space 1046; - passant 436; - passing 965; - projective 59, 785, 1273; - Q-regular 972; - r- 1002; - regular 797, 994; - secant 965; -- to k-arc 300; -- to k-cap 309; - singular 238, 797, 972, 994; - spacelike 925; - tangent 317, 436, 669, 800, 816, 965; -- to Hermitian arc 314; -- to k-arc 300; -- to k-cap 309; - thick 208, 651, 1162; - thin 435, 651; - timelike 925; - transversal 325, 805
line graph 1159
line space 68
line-diameter 437
linear algebra 14
linear combination 215
linear fractional transformations 59
linear group, general 1037
linear part of similarity 975
linear programming 857: - bound 360
linear representation 436
LINEAR SPACE 12, 70, 83, 129, 195, 198, 352, 435, 464, 465, 1003, 1147: - embeddable 129; - nondegenerate 129; - partial 70, 435; - residual 241; - restricted 131; - subspaces of 12; - with dimension 70, 87; - with dimension n 1147
linear variety 414
lineation 1280
lines orthogonal 387: - parallel 1274
linked threes 465
local diameter 437
local field 612
locally compact field 1094: - non-Archimedean 612
locally symmetric space 1111
locals 1018
logarithmic concavity 224
Lorentz space 901
Lorentz transformation 925, 935: - orthochronous 935
Lorentz--Minkowski geometry 964
Lorimer--Rahilly plane 159
Los--Tarski theorem 743
Lucas theorem 364
Lüneburg plane 163, 374, 1156
Lunelli--Sce hyperoval 122, 1199
m-covering 509, 592: - universal 593
m-homotopy 593: - elementary 593
m-spread 144: - proper 144
m-truncation 197
-plane 764
Mac Lane theorem 219
Mac Lane--Steinitz axiom 199
MacWilliams transform 371
manifold 18: - analytic 1336; - integral homology 1350; - topological n- 1276
map: adjacency preserving 620; - canonical 1088; - strong 259
mappings, equiforme 820
maps, gluing 626
Matchstick space, full 236, 262
Mathieu--Witt design 208, 212, 232, 253, 255, 268, 270, 272, 273, 275, 370, 1192, 1208, 1216
matrices 846: - congruent 846; - elementary 1038; - totally unimodular 261
matrix representation of 4-dimensional elation Laguerre planes 1348
matroid 12, 30, 195, 216, 416: - circuit 216; - cographic 216; - cycle 216; - free 208; - graphic 216; - paving 208; - polygon 216; - simple 195, 201; - uniform 208; - unimodular 216
maximal arc 313
maximal exterior set 339: - linear 339
maximal intersection, principle of 75
maximal parabolic subgroup, standard 713
maximal subspace 1169
maximal torus 694
McLaughlin graph 442, 855
MDS codes 307: - linear 308
measure function 980
meet 1120
metasymplectic space 636, 639, 696, 709, 719, 1209, 1210, 1214
metric 927
metric product 959, 968, 1004
metric space 5: - affine 974; - elliptic 990; - hyperbolic 990; - irregular 990; - pointless 1023; - projective 990; - regular 990; - singular 990
metric vector space 927, 948
minimality condition 88
Minkowski chain space, weak 827
Minkowski geometry 793, 807, 811, 828
Minkowski plane 195, 238, 239, 240, 252, 277, 746, 767, 795, 802, 814, 1295, 1327, 1328: - 4-dimensional 1342; - classical real 1342; - finite-dimensional 1340; - Miquelian 252, 770, 829, 1333; - perspectivities 767; - pseudo-pencils 767; - semiclassical 1335; - with 3-dimensional kernel 1341; - with 4-dimensional automorphism groups 1341; - with group transitive concircular points 252
Minkowski product 828, 831: - symmetric 828
Minkowski space 195
Minkowski space-time 237, 924
Minkowskian plane 952, 953
Minkowskian space 948, 963
minor 203, 259
Miquel condition 246, 821
Miquel configuration 1328
Miquel theorem 245, 999, 1004: - first 983; - second 983; - third 984
Miquelian inversive plane 369
Miquelian plane 252
Möbius 2-graph 886
Möbius function 228
Möbius geometry 792, 807, 811, 964: - characterization 823; - d-dimensional 218, 253
Möbius plane 195, 237, 238, 239, 275, 375, 746, 767, 802, 814, 964, 1147, 1194, 1327, 1328, 1336: - egglike 310, 1328; - embeddable 1337; - Miquelian 310, 820; - nonembeddable 1337; - ovoidal 310, 1328, 1337; - pencils 767; - perspectivities of 767; - semiclassical 1334; - spherical 1337; - with 2-transitive group 252; - with point-transitive group 252
Möbius space 195, 218, 253, 794, 808, 816, 895: - strongly Miquelian 824
Möbius strip 1279, 1299
Möbius-closure 997
model 742: - full 741; - infinitesimal 1304; - parabola 1287, 1292; - partial 743; - plane 242; - residual plane 240; - spear-cycle 1339; - -confined 752; - -free 754; - -open 752
modular identity 1120
modular law 1048
module 805: - faithful 805; - free 784, 1036; - left 784; - torsion free 1125; - with high weight 713
Molien--Poincaré series 869: - harmonic 870, 871
monomials standard 714
monomorphism 743
Moore geometry: generalized 463; - proper 465
Moore graph 450, 462, 1154
Moorhouse conjecture 368
Morita equivalence 1139, 1137
MORPHISM 3, 77, 78, 489: - distant-preserving 1053; - neighbour-preserving 1053; - of affine chain spaces 832; - of bimodules 1139; - of chamber systems 489; - of complexes 483; - of curve spaces 829; - of frames 1018; - of geometric lattices 255; - of geometries 77, 78; - of lattices 1136, 1139; - of models 743; - of modules 784; - of projective spaces 661, 664; - of rings 832; - of rings, surjective 1040; - of S-modules 1139; - of spaces 650; - of structured complexes 527; - of transformation groups 1028; - of V-spaces 1053; - special, see -- type preserving; - type preserving 489
Mostow's rigidity theorem 896
motion 820, 928, 975, 976, 995: - Q- 991, 996; - - 995
motion group 1304: - elliptic 1294; - Euclidean 1295; - hyperbolic 1294
Moufang blueprint 630, 632
Moufang building 585, 591, 629: - natural labelling of 630
Moufang condition 46: - topological 1309
Moufang conditions 423, 424
Moufang generalized m-gon 585
Moufang loop 40
Moufang metasymplectic space 707
Moufang oval 251
Moufang plane 153, 662, 969, 1289: - compact projective 1285
Moufang property 468
Moufang quadrangle 633, 1308
Moulton plane 11, 35, 1287, 1289, 1291, 1296, 1299, 1300
multiplicity freeness 1149, 1152
mutation 1287, 1291
n-dimensional universal model 262
n-gon, compact (connected) 1307
n vectors, equiangular 879
natural labelling 589
near n-gon 458, 460, 465: - classical 459; - ovoidal 459; - regular 460; - sporadic 459
near hexagon, regular 460
near octagon 661: - regular 461
near polygon 676
near-pencil 130
nearfield 15, 111, 151, 1288: - irregular 152; - locally compact connected 1288; - kernel 111; - regular 151
nearfield plane 1156
neighbour 1043: - dually 1043
neighbour relation 1043, 1135
neighbourhood 1158
net 465, 794: - Desarguesian 369; - double 764; - of circles 1332; - of points 1332
Netto circular space 269, 272
Netto system: classical locally 1194, 1205; - generalized 1156; - triple 1156
Neumaier geometry 1196, 1202
Newton numbers 854, 860
Niemeier lattice 910
nilpotent element 702
non-Desarguesian plane 13
nontangential inclusion 1020
norm, additive 1097
normal rational curve 298
nucleus 121, 126, 148, 298, 1000: - left 147; - middle 146; - of normal rational curve 304; - right 146
numbered complex 623: - with group action 494
numbering 484, 527: - of building 527
o-polynomial 121
obstructions to -coordinatizability 260
octahedron 871
octaves 690
octonion 690, 1263
octonion algebra 1065: - split 1066
octonion division algebra 1066
opposite plane 1274
opposition 417
optimal condition 89
orbit 868, 1028, 1281: - on DLS 269
ORDER 124, 1121, 1168, 1169: - feasible 1169; - i- 221; - in lattice 1121; - known type 1168; - of affino-projective space 234; - of Benz plane 238; - of chain space 793; - of flag 221; - of partial linear space 435; - of unital 124, 125; - partial 1018; - temporal 1018
Ore conditions 1061: - left 1054; - right 1054
Ore domain, left 1125, 1138
Ore geometry 1133
oriflamme complex 558
origami space, full 262
orthogonal decomposition 954
orthogonal sum 954
orthogonality 202, 949, 950: - of hyperbolic lines 892
orthogonality relation 988: - affine 972
orthoscheme 896: - complete 896; - doubly truncated 897; - ordinary 897; - simply truncated 897
Ostrom dimension 146
Ostrom--Foulser plane 160
Ott--Schaeffer plane 161
oval 57, 119, 120, 298, 317, 965, 1021, 1328: - closed 1310; - complete 298; - symmetric with respect to point 969; - topological 1332; - translation 251
oval plane, complex 1295
overlapping relation 1020
ovoid 124, 166, 242, 309, 323, 330, 402, 816, 965, 1328, 1337: - closed 1310; - of generalized quadrangles 403; - of polar spaces 330
p-adic number 1263
packing 327
pair: complementary 1267; - regular 395; - skew 1267
Paley 2-graph 886
Paley design, see Hadamard design
Pappian space 965
Pappus configuration 91
Pappus theorem 31, 117, 970
Pappus--Pascal theorem 969, 983
parabola 802: - skew 1292
parabolic subgroup 79, 392, 479, 564, 567, 1087, 1089, 1185: - maximal 1183; - minimal 1183; - standard 689, 707; - theorem 565
parallel 33
parallel axiom 1274
parallel classes 762, 792
parallel relation 827
PARALLELISM 40, 97, 98, 724, 1327: - in affine ring plane 1076; - in affine space 33, 724; - in block space 98; - in design 352; - in Laguerre plane 746; - in Leiß @ ner plane 1079; - in Minkowski plane 746; - in Möbius plane 746; - in nets 368; - in quasi-affine space 97; - of hyperbolic lines 892
parallelodromy condition 1079
parameter system 539, 625
parameters 548: - topological 1308
parapolar space 688, 693, 717, 719, 720
paraprojective space 654
partial flock 339
partial geometry 465: - embedded in projective space 448; - parameters 441; - proper 441
partial linear gamma space 684, 722
partial linear space 435, 650: - of order (s,t) 435; - thick 435; - thin 435
partial model: degenerate with respect to some extension process 760; - -degeneration 760
partial plane, see partial linear space
partial quadrangle 449: - thin 450
partial spread 327
partial t-spread 324: - maximal 324
partition: partial G- 213; - t- 208
partition axiom 199
partition space 224
Pascal theorem 970, 983
Pasch axiom 10, 36, 87, 447, 653, 1223, 1270, 1279
Pasch configuration 213
Pasch--Veblen axiom 68
path 520, 956, 975: - in graph 386; - reduced 520
pencil 72, 245, 300, 978, 1271, 1329: - line 1277; - of hyperplanes 304; - of points 765, 768; - tangent 762, 1329
Penrose tiling 865
pentagon 719, 857
pentagram 275
periods of time 1018
permutation group 16, 268: - n-DLS primitive 268; - n-homogeneous 268; - of quadric 815
permutation rank 728
perpendicular bisector 989
perspective 30
perspective closedness 674
perspective relation in lattices 1120
perspectivity 60, 115, 766, 1271
Petersen graph 84, 91, 440, 463, 466, 856, 890, 904, 907, 1149, 1150, 1154
Pfaffian 51
Pickert--Buekenhout theorem 969
pigeonhole principle, topological 1277
planar condition 141
planar difference sets 356
planar representation: of Laguerre plane 1343; - of Minkowski plane 1339
planar space 230: - L- 235; - locally affine 230; - locally projective 230; - sharply locally projective 230; - strongly locally affine 254; - strongly locally projective 230; - weakly locally projective 230
PLANE 816, 950: - Desarguesian L- 1080; - F-planes, connected 1070; - F-planes, transvection 1070; - in Alt(7)-geometry 1166; - in design 375; - in projective space 964; - model 244; - null 925; - of affine space 971; - of inner product space 847; - Q-admissible 816; - spacelike 925; - tangent 309; - timelike 925; - translation L- 1080
Plücker coordinates 715
Plücker relations 715
Poincaré model 893, 895
Poincaré series 540
Poincaré space 1002
POINT 6, 10, 86, 93, 109, 111, 124, 351, 606, 785, 790, 791, 793, 797, 811, 815, 835, 1021, 1022, 1026, 1294: - 3-regular 413; - 4-th Q- 968; - absolute 118, 663; - adherent to region 1021; - affine 1274; - at infinity 972, 997; - belonging to region 1019, 1021, 1022; - coaffine 1275; - cocatenal 790, 793; - collinear 387, 968; - concircular 238, 1328; - conjugate 811, 815; - dependent 958; - distant 785, 793, 797, 818, 996, 1045; - double 814, 816, 965; - elliptic 875; - Euclidean 968; - exterior hyperbolic 892; - free 1128; - Hermitian 1294; - hyperbolic 891; - ideal 848, 893; -- hyperbolic 892; - in affine ring plane 1076; - in Alt(7)-geometry 1166; - in Barbilian space 1045, 1135; - in Benz plane 1327; - in closure set 198; - in DLS 196; - in generalized n-gon 745; - in geometric lattice 1264; - in group coset geometry 395; - in Klingenberg space 1060; - in Laguerre plane 240, 746; - in lattice 1128; - in Leiß @ ner plane 1079; - in Minkowski plane 240, 746; - in Möbius plane 746; - in physical space 1026; - in projective plane 109; -- over octonion algebra 1067; - in projective space 551, 964; - in shadow space 93; - in Steiner system 748; - independent 1044, 1045; - interior 111; - isotropic 58, 1294, 1296, 1297; - near 996; - nonsingular 318; - of affine plane 744; - of affine space 971; - of chain geometry 806; - of curve space 829; - of De Sitter space 848; - of design 351; - of elliptic space 847; - of Euclidean space 849; - of finite classical polar space 330; - of generalized Hughes plane 111; - of generalized quadrangle 521; - of hyperbolic space 848; - of inner product space 847; - of k-net 745; - of Klein space 1000; - of Lie geometry 1349; - of locally classical geometry 1226; - of Minkowski plane 1339; - of 835; - of polar space 556; - of projective line 785; - of projective plane 744; - of projective space over ring 1042; - of semibiplane 86; - of space 649; - of Sperner plane 1004; - of spherical space 847; - of translation plane 141; - of ) 550; - opposite 1090; - orthogonal 387; - parallel 791, 811, 815; - Q-regular 968; - Q-singular 968; - r- 1002; - R-rational 567; - regular 395; - self-conjugate 391; - semiregular 370, 415; - simple 965; - singular 58, 318; - singular ideal 239; - singularly collinear 238; - skew-Hermitian 1294; - unimodular 1128; - with complement 1128
point graph 386, 437: - of cyclotomic type 445; - of partial geometry 441
point sets, equidistant 880
point system 1124
point-diameter 437
POLAR SPACE 45, 49, 58, 329, 455, 553, 556, 621, 649, 662, 685, 717, 719, 968, 1161, 1202, 1203: - abstract 45; - affine 724, 1195; - classical 675; - classification 58; - classification of nondegenerate 676; - dual 698; - embeddable 675; - half dual 678, 698, 719; - lower truncation of 1223; - Moufang 675, 698; - of rank n 662; - orthogonal 49; - symplectic 49; - thick-lined 1161; - thin 557; - thin from above 685; - truncation of 1222; - unitary 49
polar transvection 663, 705, 726: - group 728
polarity 43, 78, 119, 124, 562, 658, 663, 772, 968, 990: - classification 58; - fundamental 811; - induced by quadratic form 968; - nondegenerate 562; - of generalized quadrangles 404; - orthogonal 119, 124, 666; - symplectic 450, 666; - unitary 124, 666
polarization identity 923
pole 968
polyhedral group 871
polynomial: annihilator 901, 902; - harmonic 869; - homogeneous 812, 900; - minimal 813; - p- 220
Pontrjagin theorem 1263
poset 75, 485: - pure 485
ppm-space, canonical 1025
preapartment 88
pregeometry 76
primitive concepts 5, 1017
product set 849
product space 849
projection 576: - canonical 77
projection map 576
projection of quadric 319
projection pencil 765
projection pseudo-pencil 765
projection set 765: - pseudo- 765
projective n-space: topological 1265
projective closure 972
projective group of Barbilian space: full 1052; - little 1052
PROJECTIVE PLANE 11, 34, 109, 170, 385, 465, 744, 750, 1279: - almost 1275, 1298, 1300; - Archimedean 1311; - classical 389, 1270; - corresponding to metric vector space 969; - derived 1329; - Desarguesian 109, 115, 964, 1333; -- complex 1333; - free 762; - free extension process 754; - modified real 1296, 1300; - Moufang 631; - non-Desarguesian 11, 1286; - nondegenerate 754; - obtained from projective plane 34; - over Cayley algebra 572; - over Cayley octonions 1285; - over split octonions 1066; - Pappian 770, 964, 1333; - perspectivities 767; - projection pencils 767; - proper 110; - punctured 1275; - real 1300; - topological 1266, 1283, 1332; - -free constructions of 750; - -hyperfree elements in a partial 752; - -open 763
projective set 657
PROJECTIVE SPACE 9, 30, 39, 70, 195, 551, 816, 1129: - associated to projective lattice geometry 1129; - associated to vector space 964; - Desarguesian 550, 964, 967, 1155, 1261; - generalized 207; - missed 197; - of infinite dimension 1263; - over full matrix ring 1056; - over ring 1038, 1042; - over topological division ring 1260; - Pappian 964; - possibly reducible 39; - reducible 1161; - thick 551; - thin 551; - topological 1259; - which corresponds to vector space 964
projectivity 60, 115, 766, 786, 806, 968, 1271: - affine 767; - of Möbius geometry 824; - Q- 991; - - 995
proof 743
PROPERTY: A 717; - B 989; - B 233; - BB 245; - 520; - 1131; - 2 524; - 3 524; - CS1 524; - D 1130; - 520; - D1,2,3 538; - Deg 1152; - F 521; - Fth 1152; - hereditary 76, 1262; - HH 607; - IC 95; - Int-F 605; - Int-O 605; - IP 95, 1152; - 1152; - L1,L2 1004; - LH 607; - Lin 606; - LL 95, 607, 1149; - LS 95; - M 246; - M8 246; - MF 1149, 1152; - NT 1152; - O 607, 1152; - OO 95; - Pri 1152; - Q 537; - QS 1069; - R0 1124; - R2 1152; - Sph 1152; - Sph Cox 1152; - Th 1150; - TMF 1152; - U 95; - UC 1152; - X 199
pseudo-isometry 928
pseudo-modularity 220
pseudo-oval 1310, 1348: - topological 1348
pseudo-perspectivity 766
pseudo-similarity 929
pseudomanifold 504
pseudometric space, pointless 1023
pseudoquadratic form 46: - trace-valued 46
pseudoquadric 671
Pythagorean field 933
Q-conformity 999
Q-distance 990
Q-knot 968
Q-measure 999
Q-sphere 973: - proper 973
Q-symmetry 968
quad 459
quadrangle 677: - property 1137; - section condition 1069; - symplectic 1308
quadratic cone 167
quadratic set 242, 816, 965: - degenerate 816; - ruled 965, 967
quadratic space, nondegenerate 561
quadratic surface: hyperbolic 166; - elliptic 166
quadric 389, 822, 990: - affine 972; - associated with quadratic form 964; - at infinity 972, 980; - elliptic 237, 814, 1328; - hyperbolic 237, 814; - ovoidal 965; - perfect - 244; - ruled 965, 1328
quadric circle geometry 991
quadrilateral 417
quadrilaterals in perspective 417, 418
quadruple 60: - equi-anharmonic 60; - harmonic 60
quad 677
quasi-design 195, 208
quasi-inversion 250
quasifield 141: - locally compact connected 1288
quasipolarity 663: - determined by -sesquilinear form 665
quasiprojective set 657
quasisimplicity 1100
quasisplit group 568
quaternion 43, 813, 1263
quaternion algebra 636: - split 1065
quaternion division ring 673
quotient 77: - 2- 77; - of block space 67; - of geometry 77
quotient of affine polar space: improper 1196; - standard 1196, 1202, 1203
R-module flat 1136
r-net 86
R-space 499
radical 237, 567, 650, 664, 950: - of planar space 237; - solvable 1301
radius 972
Radon transform 228
RANK 30, 39, 75, 552, 662: - in lattice 1121; - of building 1308; - of chamber system 488, 678; - of complex 483; - of finite partial model 762; - of flat 277; - of free module 784, 1041; - of geometry 47; - of hyperbolic diagram 1172; - of incidence matrix 362; - of integral lattice 908; - of J-star 1185; - of lattice 39; - of parabolic subgroup 1183; - of poset 485; - of projective space 552; - of projective subspace 964; - of Q-sphere 973; - of subset 75; - of vector space 30; - polar 688; - singular 652; - stable 785, 1039; - uniform 1121
rapid growth 262
rational representation of chain geometry 831
realization of blueprint 627
reciprocity 78
rectangle, - 828
rectangle property 243
reduced expression 503
reducibility 749
reductive group 567
Ree group 466, 470
Ree unital 374, 470, 1155
Reed--Muller code 367: - punctured 367
reflection 43, 505, 699, 1282: - affine 1096; - fundamental 699; - with respect to circle C 249
reflection group 869
region: bounded 1023; - empty 1023
regions 1018, 1029: - r-approximately overlapping 1029
regular action 114
regular space 561
regular system of order m 332
regularity condition on projectivities 770
regulus 51, 97, 327, 805: - generalized 809; - opposite 51; - 454
regulus condition 419
regulus net 447
representation 769: - algebraic 256; - irreducible 769, 872; - linear 256; - of group 872; - projective 256; - rational 812; - smooth 812
residual connectedness 679
residual space 794
residual trace 794, 796
residually connected 47
RESIDUE 47, 76: - bottom 202; - of flag 76; - top 197, 202
residue geometry 20
resolution 352
restricted -spaces 254
restriction of closure space 199
retraction 527
retract, absolute neighbourhood 1350
rhomboedra 865
Riemannian manifold 1111
rigid plane 1293
rim 216
ring 1064: - alternative 1067; - biternary 1312; - chain 1060; - completely primary and uniserial 1122; - full matrix 1039, 1056; - 1076; - local 788, 791, 833, 1058, 1060; - of double numbers 960; - of dual numbers 960; - oppositive 1036; - planar sexternary 1062; - planar ternary 1062; - primitive 834; - regular 1121; - semilocal 834; - semiprimary 789, 1039; - semisimple 811; - two-sided units 1035, 1070, 1077; - valuation 1061; - Von Neuman regular 1134; - weakly quadratic 822
Robertson--Walker space-time 930, 937
roof 276
root 506, 567, 583, 699: - affine 1096; - in building 583; - opposite 508
root group 584, 703, 727: - exceptional 728
root group space 708: - exceptional 720
root system 18, 699, 854, 861, 870, 888, 1096: - relative 1089; - simple 19
rotation 979
s-property, see splitting property 1137
Sandler semifield 154
satisfaction relation 742
scalar: spacelike 934; - timelike 934
scalar product 1111
Schläfli configuration 855
Schläfli graph 856, 905
Scum theorem 259
secant 300: - m- 436
secant segments theorem 984
second main characterization of buildings 598
section 994: - affine-dual-affine 1214
segment, straight 1021
Segre manifold 811
Segre product 831
Segre theorem 57, 247
Segre variety 325
semi-affine plane 235
semi-affinity 929
semi-building 91
semi-oval 242, 317: - regular 317
semi-ovoid 242, 317
semibiplane 86, 217, 1147
semidirect product, almost 1301
semifield 152, 1287
semifield plane 152, 1291: - proper 153
semilinear space, see partial linear space
semilinear transformation, -semilinear 41
semipartial geometry 403, 448: - proper 449
semisimple algebraic group 567
semisimple group 567
sentence 742
separateness relation 1021
separation 927: - k- 206
set 808, 811: - abstractive 1020; - affinely dependent 971; - affinely independent 971; - closed 198; -- under reguli 808; -- with respect to r-distance lines 811; - concordant 228; - dependent 200; - few-distance 888; - independent 200, 964, 994; - of odd type 320; - of type 316, 318; - orthogonal 886; - partially ordered 75; - quadratic 1331; - representative 1021; - skew characteristic 263; - spanning 200; - standard 835; - symplectic 886
set of permutations, geometric 277
set of representatives 323, 324: - minimal 324; - reduced, see -- minimal
Seymour decomposition theorem 261
shadow 93, 499: - geometry 615; - poset 615; - space 93, 604, 682; -- embeddings of 712; -- over j 682
shape 1026
shear 173: - plane 1274
shellability 589
shift plane 1287, 1292
signature 928: - of inner product 846; - of integral lattice 908; - of symmetric matrix 846
similarity 820, 928, 975, 977, 995: - of Minkowskian plane 979
simple connectedness, residual 77
simple design 351
simplex 42, 483: - empty 483; - hyperbolic 895, 896; - opposite 580; - regular 867
Singer group 114
singular space 318
singular stroke 1148
singular subspace 444
size: i- 221; - of i-variety 221
skew affine space 97
skew field, see division ring
skew-hyperbolic plane, quaternion 1295
Skolemization 773
sliced space 1268
Smith graph 866
Smith normal form 363
socle 181
space 6, 649, 650: - complete 650; - LP- 764; - m-closure 216; - nondegenerate 650; - paraprojective 684; - rich n- 1265; - singular 650; - stable n- 1265; - strongly parapolar 688; - topological n- 1265; - with homomorphism 1058
spanning diamond 1123
spanning family 1120
Sperner space 98, 795
sphere: algebraic n- 811; - unit 901
spherical design 863
spherical harmonics, zonal 852
spherical space 847, 1002
spin group 635
spinor 51, 52: - pure 52
split group 568
splitting property 1137
spoke 216
sporadic group 1243
spread 124, 143, 327, 330, 402, 1288: - of generalized quadrangle 403; - partial 1274; - planar regular 807; - proper 143; - regular 124; - subregular of index k 327
square classes 1000
square ratio 979
stability 1265
stabilizer 17, 1281: - lemma 1340, 1344; - of quadrangles 1290
stable plane 1258, 1266, 1270: - homogeneous 1293; - locally compact 1273; - locally connected 1273; - proper 1271
stable space, sliced 1269
standard representation of Minkowski plane 1340
star 483, 1173
Steinberg relations 703
Steinberg representation 1092, 1094
STEINER SYSTEM 13, 56, 124, 208, 351, 353, 373, 1149: - 3-(22,6,1) 908; - 4-(23,7,1) 903; - 5-(24,8,1) 854, 860, 887; - free extension process 756; - - 748; - nondegenerate - 756; - partial - 748; - quadruple 40, 230, 1192; - triple 370; - -free constructions of - 752
stereographic projection 241, 816, 817, 893, 910, 993, 1327, 1337
stiffness 1269, 1281
Stirling number 224
Stokes law 882
strain 173
Strambach's -plane 1297, 1299
strength: of measure 874; - of spherical design 863
string 486, 1173
strong B-group 179
strong exchange axiom 198
strong regularity 360
strong transitivity 563
structure 4: - - 742
structure constants 701, 803
structured complex 526
sub-n-gon, proper 405
subalgebra 698
subcomplex 574: - convex 574
subdiagram 1152
subgeometry 76, 204, 259: - induced 76; - of x in G 1129; - -free 761
subgroup 1104: - anisotropic 571; - bounded 1097; - maximal arithmetic 1110; - maximal parabolic 20; - nonsemi-simple 20; - parabolic hyperspecial 1110; - parahoric 1101; - planar 1303; - quasiperspective 1303; - S-arithmetic 1104; - t-homogeneous 869; - torsion-free 1106
submanifolds, isoparametric 1309
submodel 749, 774: - -closed 774; - -hyperfree 752
suboctagon 407
subquadrangle 405
subregular sequence of spreads of length k 327
subring: admissible 1054, 1055; - maximal for morphism 1055
subset 216, 219, 576, 598, 1120: - algebraically independent over F 219; - determining Barbilian domain 1075; - flat 533; - gated 576; - geodetically closed 459; - independent 552; - m-closed 216; - prelinear 557; - proper 965; - proper m-closed 216; - regular 1019; - singular 650; - spherical 598
SUBSPACE 36, 45, 70, 71, 73, 663, 676, 821: - absolute 663; - adjacent 676; - affine 9, 971, 975; - complementary 950; - distant 1043; - dually distant 1043; - generated by subset 650; - geodetically closed 684; - in projective space 551; - irregular 951; - isoclinic 897; - isotropic 951; - linear 70; - maximal singular 952, 954; - maximal totally isotropic 330; - nonsingular 994; - of chain space 821; - of closure space 73; - of free module 1037; - of hyperplane space 71; - of line space 70; - of point-line space 650; - of polar space 556; - of projective space 964; - of projective space over ring 1042; - of ) 550; - open 1267; - parallel 971; - planar 231; - projective 36, 964; - regular 951; - singular 70, 951, 952, 994; - spanned by vectors 949; - totally isotropic 45, 70, 561; - totally singular 45; - transversal 1043
substructure, - 742
subtorus 1087
Suetake plane 165
sum 551: - of flats 1048; - orthogonal 851
supersum of DLS's 204
support 773
Süss group 820
Suzuki chain 1224
Suzuki group 54
Suzuki--Tits ovoid 54, 244, 469
Suzuki--Tits plane 252
switching of graphs 878
symmetric group 3
symmetric plane 1280, 1298, 1304: - generalized 1306; - k- 1305
symmetric space: generalized 1305; - isotropic 1297; - k- 1305; - Riemannian 1309
symmetry 420, 1304: - affine Q- 976; - Q- 957
symplectic polarity 390
symplecton 636, 688, 1170
syntheme 401
system of apartments 90, 526
system of blocks of imprimitivity 99
system of groups 566: - satured 566
system of large apartments 90
system of regular apartments 92
t-design 13, 351: - spherical 868
t-design code 371
t-property, see triangle property
t-regulus 325
t-spread 324: - regular 324
t-wise balanced design 352
-blocking set 321, 323: - minimal 323; - reduced, see -- minimal
tactical decomposition 352
tail 1173
Tamagawa numbers 1109, 1110
tangent 300, 973, 1328: - plane 1328; - to 318
tangent bundle plane 1071
tangent curve 305
tangent space 421, 811, 816, 1071, 1073: - to Veldkamp plane 1071, 1073
tangent theorem 983
tangent vector 1071, 1073: - to Veldkamp plane 1071, 1073
Teirlinck theorem 210
term 742
ternary field 1284: - topological 1271
tetrahedron 871: - hyperbolic 895
theory 743: - closed 743; - equivalent 743; - inductive 743; - reproducible 760; - universal 743
thick support 1162
thickness 387
tiling: aperiodic 865; - icosahedron 865
Timmesfeld diagram 1174, 1178
Tits generalized quadrangle 393
Tits group 709
Tits ovoid 310, 370
Tits property 468
Tits system 479, 549, 563, 572, 688, 1098: - saturated 564; - split 591
Tits theorem 967
Tits--Weiss theorem 1307
top heaviness 224
topology 1025: - compact-open 1280; - Lie group 1111; - product 1090; - quotient 1088
torus: k-split 567, 1087; - maximal 567; - split 1087
totally singular subspaces, maximal 330
trace set 665
transformation, semilinear 929
transitivity 109: - 3-
allel 825: - F- 251; - of Q-symmetries 977
transitivity theorem 976, 987
translation 141, 250, 798, 975, 1078, 1291: - group 43, 421; - hyperoval 121; - - 1079; - ovoid 244; - plane 141, 334, 369, 770, 988, 1291, 1337; -- affine 1156, 1305; -- almost projective 1302; -- non-Desarguesian 1306; -- nontopological 1288; -- tangent 1304; -- topological 162; - vector 975
translation complement 145: - linear 145
translation generalized quadrangle 420, 421, 423
transvection 42, 726: - central 1049; - group, h- 1051; - h- 1051
trial solid 391
triality 53, 78, 391, 690
triangle axiom 1231
triangle, ideal 894
triangle property 887, 1136
triangular lattice 865
triple: balanced 1128; - directly balanced 1128; - flat 880; - odd 880
truncated icosahedron 869
truncation: 1201; - hyperplane- 209; - lower 1221, 1222; - n- 202; - point- 209; - point hyperplane- 201; - upper 202
truncational multiplicity freeness 1152
Tutte group 260
Tutte homotopy theorem 260
twin buildings 591
twin reflections 1302
twisted cubic 161
twisted field, generalized 154
two-distance set, spherical 854
type 21, 47, 75: - distinguished 569; - of gallery 484; - of reality 836
type function 75
type set, opposite 581
U-property 1131
unfolding 1162, 1226
uniform element 1121
uniformity 1027
unimodular element 1037
unisecant 318
unit vector 1075
unital 124, 125, 358, 452, 886: - classical 124; - Hermitian 273
universal part of theory 743
universe 742
V-plane 1051: - Desarguesian 1051; - transvection 1051
valency 438
Vámos DLS 256
Van Lint--Schrijver geometry 445
variable 742
variety 208, 212: - Hermitian 242; - improper 196; - in DLS 196; - modular 212; - projective 811; - proper 196; - quasiprojective 810, 831; - thin 208
Veblen axiom, see Pasch axiom
Veblen--Wedderburn system 141
Veblen--Young axiom 48, 551, 1117, 1129
Veblen--Young theorem 36, 229, 653
vector 846: - 4-th Q- 958; - balanced 864; - isotropic 846; - negative 846; - null 927; - of minimum norm 916; - orthogonal 927; - positive 846
vector space 4: - anisotropic 927; - nonsingular 927
Veldkamp plane 1036
Veldkamp space 656, 1036, 1045, 1135: - affine 1050; - -transitive 1050; - dual affine 1050; - dually -transitive 1050
Veronese manifold 801, 836
vertex 318, 483: - of cone 318
Von Neumann geometry 1134
Von Staudt chain 837
Von Staudt--Hua theorem 834
Walker plane 161, 162
Walker quasifield 162
wall 506
weak ordering 517
Weetman theorems 1158
weight 371, 700, 712, 859: - antidominant 712; - dominant 712; - essential 371; - fundamental 712; - minuscule 714
weight enumerator 540, 548
Weil conjecture 1109, 1110
Weil restriction 1104
Weyl group 385, 505, 568, 700, 870, 871: - absolute 568; - affine 1096, 1098; - relative 1089
Weyl homotopy 525, 596: - elementary 525
wheel 216
whirl 216
Whitney number 224
Wille geometry 195: - of grade n 218
Witt decomposition 956
Witt decomposition theorem 954
Witt design, see Mathieu--Witt design
Witt index 363, 666, 927, 954, 967
Witt lattice 854, 861
Witt ring 1311
Witt space 1202
Witt theorem 562, 689
Witt--Bose--Shrikhande space 1155
word 307, 484, 769: - reduced 769
Zeeman theorem 935
zero divisor 1063: - right 1132
zonotope, space-filling 261
space, proper 458
-brick 750, 757
-construction 749
-free construction 750: - compatible with respect to homomorphism 758; - tied 757
-Hermitian form 559, 633: - associated to -quadratic form 561
-quadratic form 561