189:
The table below shows all possible nontrivial linear spaces of five points. Because any two points are always incident with one line, the lines being incident with only two points are not drawn, by convention. The trivial case is simply a line through five points.
40:
with the line. Each two points are in a line, and any two lines may have no more than one point in common. Intuitively, this rule can be visualized as the property that two straight lines never intersect more than once.
344:
398:
268:
93:, where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line.
193:
In the first illustration, the ten lines connecting the ten pairs of points are not drawn. In the second illustration, seven lines connecting seven pairs of points are not drawn.
215:
201:
222:
208:
88:
287:
101:
293:
166:
Some authors drop (L3) when defining linear spaces. In such a situation the linear spaces complying to (L3) are considered as
464:
349:
186:
with its points and lines constitutes a linear space, moreover all affine and projective spaces are linear spaces as well.
509:
487:
49:
532:
475:
55:
434:
257:
8:
129:
429:
21:
505:
483:
460:
452:
419:
45:
183:
456:
526:
493:
497:
424:
409:
267:
90:
414:
104:
in 1964, though many results about linear spaces are much older.
221:
214:
207:
200:
339:{\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})}
33:
155:(L2) every line is incident to at least two distinct points.
152:(L1) two distinct points are incident with exactly one line.
24:. A linear space consists of a set of elements called
346:
which is not a single point or a single line, we have
393:{\displaystyle |{\mathcal {P}}|\leq |{\mathcal {L}}|}
352:
296:
58:
36:
of the points. The points in a line are said to be
392:
338:
82:
44:Linear spaces can be seen as a generalization of
524:
519:. Cambridge University Press, Cambridge, 1992.
248:points containing a line being incident with
265:
219:
212:
205:
198:
252: − 1 points is called a
480:Einführung in die endliche Geometrie II
525:
515:L. M. Batten, Albrecht Beutelspacher:
290:shows that in any finite linear space
136:are called points and the elements of
446:
162:contains at least two distinct lines.
504:. Cambridge University Press, 1992,
482:. Bibliographisches Institut, 1983,
148:if the following three axioms hold:
328:
13:
517:The Theory of Finite Linear Spaces
380:
360:
318:
308:
14:
544:
266:
220:
213:
206:
199:
262:
195:
28:, and a set of elements called
386:
374:
366:
354:
333:
303:
77:
59:
1:
440:
281:
107:
274:
237:
234:
231:
228:
132:, for which the elements of
7:
403:
275:near pencil with 10 points
177:
32:. Each line is a distinct
10:
549:
451:, Universitext, Springer,
170:and those that do not are
502:A Course in Combinatorics
457:10.1007/978-3-642-15627-4
447:Shult, Ernest E. (2011),
52:, and more broadly, of 2-
20:is a basic structure in
288:De Bruijn–Erdős theorem
83:{\displaystyle (v,k,1)}
476:Albrecht Beutelspacher
394:
340:
84:
395:
341:
85:
435:Partial linear space
350:
294:
56:
140:are called lines.
130:incidence structure
533:Incidence geometry
430:Molecular geometry
390:
336:
244:A linear space of
80:
22:incidence geometry
490:, p. 159 (German)
466:978-3-642-15626-7
330:
279:
278:
242:
241:
540:
469:
449:Points and Lines
420:Projective space
399:
397:
396:
391:
389:
384:
383:
377:
369:
364:
363:
357:
345:
343:
342:
337:
332:
331:
322:
321:
312:
311:
270:
263:
224:
217:
210:
203:
196:
89:
87:
86:
81:
548:
547:
543:
542:
541:
539:
538:
537:
523:
522:
467:
443:
406:
385:
379:
378:
373:
365:
359:
358:
353:
351:
348:
347:
327:
326:
317:
316:
307:
306:
295:
292:
291:
284:
184:Euclidean plane
180:
110:
57:
54:
53:
12:
11:
5:
546:
536:
535:
521:
520:
513:
494:J. H. van Lint
491:
472:
471:
465:
442:
439:
438:
437:
432:
427:
422:
417:
412:
405:
402:
388:
382:
376:
372:
368:
362:
356:
335:
325:
320:
315:
310:
305:
302:
299:
283:
280:
277:
276:
272:
271:
240:
239:
236:
233:
230:
226:
225:
218:
211:
204:
179:
176:
164:
163:
156:
153:
109:
106:
100:was coined by
79:
76:
73:
70:
67:
64:
61:
9:
6:
4:
3:
2:
545:
534:
531:
530:
528:
518:
514:
511:
510:0-521-42260-4
507:
503:
499:
495:
492:
489:
488:3-411-01648-5
485:
481:
477:
474:
473:
468:
462:
458:
454:
450:
445:
444:
436:
433:
431:
428:
426:
423:
421:
418:
416:
413:
411:
408:
407:
401:
370:
323:
313:
300:
297:
289:
273:
269:
264:
261:
259:
255:
251:
247:
227:
223:
216:
209:
202:
197:
194:
191:
187:
185:
175:
173:
169:
161:
157:
154:
151:
150:
149:
147:
143:
139:
135:
131:
127:
123:
119:
115:
105:
103:
99:
94:
92:
91:block designs
74:
71:
68:
65:
62:
51:
50:affine planes
47:
42:
39:
35:
31:
27:
23:
19:
516:
501:
498:R. M. Wilson
479:
448:
425:Affine space
410:Block design
285:
253:
249:
245:
243:
192:
188:
182:The regular
181:
171:
167:
165:
159:
146:linear space
145:
141:
137:
133:
125:
121:
117:
113:
111:
98:linear space
97:
95:
43:
37:
29:
25:
18:linear space
17:
15:
254:near pencil
102:Paul Libois
441:References
415:Fano plane
282:Properties
168:nontrivial
108:Definition
46:projective
371:≤
229:10 lines
96:The term
527:Category
512:. p. 188
404:See also
238:5 lines
235:6 lines
232:8 lines
178:Examples
128:) be an
38:incident
256:. (See
172:trivial
508:
486:
463:
258:pencil
34:subset
26:points
158:(L3)
144:is a
30:lines
506:ISBN
484:ISBN
461:ISBN
286:The
112:Let
48:and
453:doi
400:.
116:= (
529::
500::
496:,
478::
459:,
260:)
174:.
124:,
120:,
16:A
470:.
455::
387:|
381:L
375:|
367:|
361:P
355:|
334:)
329:I
324:,
319:L
314:,
309:P
304:(
301:=
298:S
250:n
246:n
160:L
142:L
138:G
134:P
126:I
122:G
118:P
114:L
78:)
75:1
72:,
69:k
66:,
63:v
60:(
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