A categorical approach to loops, neardomains and nearfields

A categorical approach to loops, neardomains and nearfields (Joint work with Philippe Cara and Tina Vervloet).

Loops, nearrings and nearfields are structures in algebra which generalize groups, rings and fields, respectively.
They may be less well-known but are very useful in areas like affine and projective geometry, coding theory
and cryptography. Neardomains (introduced by Karzel in 1965, and more broadly published in [2] below) are a
weakening of nearfields in which the associativity of the addition is further relaxed.

In this talk we focus on the categories of loops, of nearfields and of neardomains (with the obvious morphisms)
respectively. For these three categories we construct equivalent categories which involve only sets of permutations
or groups of permutations with extra properties.

In particular, the category of neardomains is equivalent to the category of sharply 2-transitive groups, and this
equivalence nicely restricts to an equivalence of the category of nearfields to the category of sharply 2-transitive
groups with an extra property.

All presently known neardomains turn out to be nearfields ! We hope that this categorical translation will help to find
(after more than 40 years) a proper neardomain, i.e. a neardomain which is not a nearfield.

References

[1] J.R. Clay: Nearrings. Geneses and applications. Oxford University Press, New York 1992.
[2] H. Karzel: Zusammenh¨ange zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und
2-Strukturen mit Rechteckaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191-206.
[3] H. Kiechle: Theory of K-loops. Lecture notes in Mathematics 1778. Springer Verlag, Berlin 2002.
[4] H. Wähling: Theorie der Fastkörper. Thales Verlag, Essen 1987.


Vrije Universiteit Brussel
[email protected]