Structure theory.

G. Wendt showed that every finite planar near-ring occurs, up to isomorphism, as a subnear-ring of a centralizer near-ring [#!We:CRFP!#]. Conversely, minimal left ideals of $2$-primitive near-rings are planar near-rings. More generally, any near-ring with a multiplicative right identity can be described as centralizer near-ring in which the multiplication is not composition of functions but a so-called sandwich multiplication [#!We:PNSN!#]. This description is one of the most interesting structural results obtained since it provides a new method to study near-rings with right identity more closely (for example: primitive near-rings, planar near-rings). The question whether a near-ring has a multiplicative right identity or not was the motivation to investigate near-rings with the focus on their multiplicative semigroups in [#!We:OTMS!#]. In this area, G. Wendt also obtained general results on zero divisors in near-rings [#!We:OTNO!#], on the multiplicative behaviour of near-rings in general, and on the multiplicative semigroup of subnear-rings of 1-primitive near-rings [#!We:LII1!#].

A different approach yielded a description of the maximal near-rings of functions on finite groups. For a given finite group $G$, it is now possible to determine whether a given set of functions $S$ on $G$ generates the near-ring $M(G)$ of all functions on $G$ by checking whether $S$ is contained in any of the maximal subnear-rings of $M(G)$. This result was obtained by E. Aichinger et al. using a combination of near-ring theory, clone theory, and group theory [#!AM:CFCN!#]. As a consequence, they give a bound for the probability $p_G$ that a randomly chosen bijection on $G$ generates all of $M(G)$ for $\vert G\vert > 5$:

$$p_G > (\vert G\vert-6)/\vert G\vert.$$

For $\vert G\vert \rightarrow \infty$, $p_G$ tends to $1$; in particular, this shows that on groups with at least $6$ elements there always exists a single bijection that generates all of $M(G)$. Depending on I. Rosenbergs famous characterization of maximal clones, this result is in line with a growing number of recent results on the probability that a randomly chosen element generates a given algebra (group,...). In his diploma thesis, C. Neumaier independently characterized the maximal subnear-rings of the zero-symmetric functions $M_0(G)$. By a careful analysis of the involutions in $G$, he obtained generating probabilities in this case [#!Ne:TMSO!#,#!Ne:TFOT!#].

A third line of research on the structure of planar near-rings was positioned around a question that was dubbed ``one of the most long-standing and probably most difficult in the vicinity of near-ring theory'' by H. Kiechle in one of the main talks at the international conference Near-rings and Near-fields 2005: Is every near-domain a near-field? Equivalently, does every sharply $2$-transitive permutation group have a regular normal subgroup? Certain finiteness conditions are known to guarantee the existence of a regular normal subgroup. P. Mayr proved that sharply $2$-transitive permutation groups whose point stabilizer has exponent $3$ or $6$ have such a regular normal subgroup [#!Ma:S2TG!#]. Apart from near-ring theory, this result is of interest for classical group theory. As consequences, certain near-domains are in fact finite planar near-fields, the varieties of zero-symmetric right near-rings with left identities satisfying $x^4 = x$ or $x^7 = x$ are varieties of commutative rings.