| Title:
|
One configurational characterization of Ostrom nets (English) |
| Author:
|
Baštinec, Jaromír |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
116 |
| Issue:
|
2 |
| Year:
|
1991 |
| Pages:
|
132-147 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Bz the quadrileteral condition in a given net there is meant the following implication: If $A_1, A_2, A_3,A-4$ are arbitrary points, no three of them lie on the same line, with coll $(A_iA_j)$ (collinearity) for any five from six couples $\{i,j\}$ then there follows the collinearity coll $(A_kA_l)$ for the remaining couple $\{k,l\}$.
In the article there is proved the every net satisfying the preceding configuration condition is necessarity the Ostrom net (i.e., the net over a field). Conversely, every Ostrom net satisfies the above configuration condition. (English) |
| Keyword:
|
net |
| Keyword:
|
Ostrom net |
| Keyword:
|
quadrilateral closure condition |
| Keyword:
|
skew field |
| Keyword:
|
quadrangular condition |
| MSC:
|
51A20 |
| MSC:
|
51A25 |
| idZBL:
|
Zbl 0737.51002 |
| idMR:
|
MR1111998 |
| DOI:
|
10.21136/MB.1991.126140 |
| . |
| Date available:
|
2009-09-24T20:44:09Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126140 |
| . |
| Reference:
|
[1a] V. D. Belousov: Algebraic nets and quasigroups.(in Russian), Kishinev 1971. |
| Reference:
|
[1b] V. D. Belousov: On closure condition in K-nets.(in Russian), Mat. Issl. G, No 3 (21), 33-44. MR 0299710 |
| Reference:
|
[1c] V. D. Belousov G. B. Beljavskaja: Interrelations of some closure conditions in nets.(in Russian), Izv. Ak. Nauk Mold. SSR 2 (1974), 44- 51. MR 0360315 |
| Reference:
|
[1d] V. D. Belousov: Configurations in algebraic nets.(in Russian), Kishinev 1979. MR 0544665 |
| Reference:
|
[2a] V. Havel: Kleine Desargues-Bedingung in Geweben.Časopis pěst. mat. 102 (1977), 144-165. Zbl 0359.50024, MR 0500497 |
| Reference:
|
[2b] V. Havel: General nets and their associated groupoids.Prac. Symp. "n-ary Structures", Skopje 1972, 229-241. MR 0735655 |
| Reference:
|
[3] J. Kadleček: Closure conditions in the nets.Comm. Math. Univ. Car. 19 (1978), 119-133. MR 0492374 |
| Reference:
|
[4] T. G. Ostrom: Vector spaces and construction of finite projective planes.Arch Math. 19 (1968), 1-25. Zbl 0153.49801, MR 0226492, 10.1007/BF01898796 |
| Reference:
|
[5] H. Thiele: Gewebe, deren Ternärkörper aus einem Vektorraum hervorgeht.Mitt. Math. Sem. Giessen, Nr. 140 (1979), 32-79. Zbl 0406.51002, MR 0542564 |
| . |
