SandwichCentralizerNearrings

1. <b>Theorem.</b> <i>Let $N$ be a planar nearring and $(N,\Phi)$ the corresponding Ferrero pair. Let $f:N\to N$. Then $f$ is an $N$-endomorphism of $N$ if and only if $f\in \Phi$.</i>

2. (Old definition from Wendt) Take the following objects:

• a group $(N,+)$,
• a nonempty set $X$,
• a mapping $\phi:N\to X$,
• a subgroup $C$ of the permutation group $S_X$,
• a homomorphism $\psi:C\to \rm{aut}(N,+)$.

Consider the set $M=M(\phi,\psi,X,N)$ of all mappings $f:X\to N$ with the property that $f\circ\sigma =\psi(\sigma)\circ f$ for all $\sigma\in C$. If $\phi$ and $\psi$ satisfy $\phi\circ\psi(\sigma)=\sigma\circ\phi$ for all $\sigma\in C$, then we can have a multiplication $*$ on $M$ given by $f*g=f\circ \phi\circ g$ for all $f,g\in M$, and $(M,+,*)$ is a nearring called a sandwich centralizer nearring.

3. <b>Theorem</b> (Wendt, old version). <i>A nearring $M$ is planar if and only if $M$ is isomorphic to a sandwich centralizer nearring $M(\phi,1,X,N)$.</i>

4. <b>Theorem</b> (Wendt, old version). <i>Let $M_1=M(\phi_1,\psi_1,X_1,N_1)$ and $M_2=M(\phi_2,\psi_2,X_2,N_2)$ be two sandwich centralizer nearrings. Let $F:N_1\to N_2$. Then $F$ is an isomorphism of the nearrings $M_1$ and $M_2$ if and only if there exist a group isomorphism $\theta:N_1\to N_2$ and a bijection $\lambda:X_1\to X_2$ with $\theta\circ\phi_1=\phi_2\circ\lambda$, and $\sigma\circ F(f)=F(f)\circ\sigma$ for all $\sigma\in C_2$, the domain of $\psi_2$.</i>

<b>Exercise.</b> Describe the orbit representatives of $M(\phi,\psi,X,N)$.