# Coordinate theorems for affine Hjelmslev planes

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## Summary

It is shown that an affine Hjelmslev plane ℋ is a translation plane if and only if each of its coordinate biternary rings B=〈k, T, T0, 0, 1〉 are linear. Addition and multiplication in the ternary ring 〈k, T, 0, 1〉 are defined by a+b=T(a, 1, b) and a·b= =T(a, b, 0), respectively, and it is proved that every biternary ring of a translation plane has the additional properties that 〈k,+〉 is an abelian group 〈k, +, ·〉 is right distributive, and T(a, 1, b)=T0(a, 1, b). Moreover, if a single linear biternary ring of ℋ has these three properties, then ℋ is a translation plane. It is shown that a translation plane is Desarguesian if and only if it has a linear biternary ring such that T=T0 and 〈k, +, ·〉 is an affine Hjelmslev ring. Hessenberg’s theorem for affine Hjelmslev planes is proved, and a special configurational condition which is equivalent to the commutativity of multiplication in each biternary ring is introduced.

## References

1. 

E. Artin,Coordinates in affine geometry, Rep. Math. Coll.,2 (2) (1940), pp. 15–20.

2. 

E. Artin,Geometric algebra, Interscience Publishers Inc., New York (1957).

3. 

B. Artmann,Uniforme Hjelmslev-Ebenen und Modulare Verbände, Math. Z.,111 (1969), pp. 15–45.

4. 

P. Y. Bacon,Coordinatized H-planes, Ph. D. thesis, Univ. of Florida (1974).

5. 

V. K. Cyganova,An H-ternar of the Hjelmslev affine plane (Russian), Smolensk. Gos. Ped. Inst. Učen. Zap.,18 pp. 44–69.

6. 

D. A. Drake,Coordinatization of H-planes by H-modules, Math. Z.,115 (1970), pp. 79–103.

7. 

G. Grätzer,Universal algebra, D. Van Nostrand Company Inc., New York (1960).

8. 

M. Hall,Projective planes, Trans. Amer. Math. Soc.,54 (1943), pp. 229–277.

9. 

W. Klingenberg,Beziehungen zwischen einigen affinen Schliessungssätzen, Abh. Math. Sem. Univ. Hamburg,18 (1952), pp. 120–143.

10. 

W. Klingenberg,Projecktive und affine Ebenen mit Nachbarelementen, Math. Z.,60 (1954), pp. 384–406.

11. 

W. Klingenberg,Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg,20 (1955), pp. 97–111.

12. 

J. W. Lorimer - N. D. Lane,Desarguesian affine H 6 elmslev planes, to appear in J. Reine Angew. Math.

13. 

J. W. Lorimer - N. D. Lane,Desarguesian affine H 6 elmslev planes, Mc Master Univ. Math. Report 55 (1973).

14. 

H. Lüneburg,Affine Hjelmslev-Ebenen mit transitiver Translationgruppe, Math. Z.,79 (1962), pp. 260–288.

15. 

E. Sperner,Affine Räume mit schwacher Incidenz und zugehörige algebraische Strukturen, J. Reine Angew. Math.,204 (1960), pp. 205–215.

## Author information

Entrata in Redazione il 16 ottobre 1973.

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Reprints and Permissions

Lorimer, J.W. Coordinate theorems for affine Hjelmslev planes. Annali di Matematica 105, 171–190 (1975). https://doi.org/10.1007/BF02414928