One configurational characterization of Ostrom nets

Jaromír Baštinec

Mathematica Bohemica (1991)

  • Volume: 116, Issue: 2, page 132-147
  • ISSN: 0862-7959

Abstract

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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 i A j ) (collinearity) for any five from six couples { i , j } then there follows the collinearity coll ( A k A 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.

How to cite

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Baštinec, Jaromír. "One configurational characterization of Ostrom nets." Mathematica Bohemica 116.2 (1991): 132-147. <http://eudml.org/doc/29294>.

@article{Baštinec1991,
abstract = {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 $\lbrace i,j\rbrace $ then there follows the collinearity coll $(A_kA_l)$ for the remaining couple $\lbrace k,l\rbrace $. 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.},
author = {Baštinec, Jaromír},
journal = {Mathematica Bohemica},
keywords = {net; Ostrom net; quadrilateral closure condition; skew field; quadrangular condition; net; skew field; quadrangular condition},
language = {eng},
number = {2},
pages = {132-147},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {One configurational characterization of Ostrom nets},
url = {http://eudml.org/doc/29294},
volume = {116},
year = {1991},
}

TY - JOUR
AU - Baštinec, Jaromír
TI - One configurational characterization of Ostrom nets
JO - Mathematica Bohemica
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 116
IS - 2
SP - 132
EP - 147
AB - 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 $\lbrace i,j\rbrace $ then there follows the collinearity coll $(A_kA_l)$ for the remaining couple $\lbrace k,l\rbrace $. 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.
LA - eng
KW - net; Ostrom net; quadrilateral closure condition; skew field; quadrangular condition; net; skew field; quadrangular condition
UR - http://eudml.org/doc/29294
ER -

References

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  1. V. D. Belousov, Algebraic nets and quasigroups, (in Russian), Kishinev 1971. (1971) 
  2. V. D. Belousov, On closure condition in K-nets, (in Russian), Mat. Issl. G, No 3 (21), 33-44. MR0299710
  3. V. D. Belousov G. B. Beljavskaja, Interrelations of some closure conditions in nets, (in Russian), Izv. Ak. Nauk Mold. SSR 2 (1974), 44- 51. (1974) MR0360315
  4. V. D. Belousov, Configurations in algebraic nets, (in Russian), Kishinev 1979. (1979) MR0544665
  5. V. Havel, Kleine Desargues-Bedingung in Geweben, Časopis pěst. mat. 102 (1977), 144-165. (1977) Zbl0359.50024MR0500497
  6. V. Havel, General nets and their associated groupoids, Prac. Symp. "n-ary Structures", Skopje 1972, 229-241. (1972) MR0735655
  7. J. Kadleček, Closure conditions in the nets, Comm. Math. Univ. Car. 19 (1978), 119-133. (1978) MR0492374
  8. T. G. Ostrom, 10.1007/BF01898796, Arch Math. 19 (1968), 1-25. (1968) Zbl0153.49801MR0226492DOI10.1007/BF01898796
  9. H. Thiele, Gewebe, deren Ternärkörper aus einem Vektorraum hervorgeht, Mitt. Math. Sem. Giessen, Nr. 140 (1979), 32-79. (1979) Zbl0406.51002MR0542564

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