The structure of $n$-uniform translation Hjelmslev planes

Abstract:

Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If $n > 1$, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane $\mathfrak {A}$ with the following property: for each point P of $\mathfrak {A}$, the substructure $^{n - 1}P$ of all neighbor points of P is an $(n - 1)$-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods $^1P \subset {\;^2}P \subset \cdots \subset {\;^n}P = \mathfrak {A}$. For $i < n,{\;^i}P$ is an i-uniform affine Hjelmslev plane under the induced incidence relation (for some parallel relation). Hjelmslev planes are called strongly n-uniform if they are n-uniform and possess one additional property; the additional property is designed to assure that the planes have epimorphic images which are strongly $(n - 1)$-uniform. Henceforth, assume that $\mathfrak {A}$ is a strongly n-uniform translation (affine) Hjelmslev plane. Let ${{(^i}P)^ \ast }$ denote the incidence structure $^iP$ together with the parallel relation induced therein by the parallel relation holding in $\mathfrak {A}$. Then for all positive integers $i \leq n$ and all points P and Q of $\mathfrak {A}$, ${{(^i}P)^ \ast }$ and ${{(^i}Q)^\ast }$ are isomorphic strongly i-uniform translation Hjelmslev planes. Let $^i\mathfrak {A}$ denote this common i-uniform plane; ${{(^i}\mathfrak {A})_j}$, denote the “quotient” of $^i\mathfrak {A}$ modulo $^j\mathfrak {A}$. The invariant $r = {p^x}$ of $\mathfrak {A}$ is the order of the ordinary translation plane ${{(^n}\mathfrak {A})_{n - 1}}$. Then the translation group of $\mathfrak {A}$ is an abelian group with 2xk cyclic summands, k an integer $\leq n$; one calls k the width of $\mathfrak {A}$. If $0 \leq j < i \leq n$, then ${{(^i}\mathfrak {A})_j}$ is a strongly $(i - j)$-uniform translation Hjelmslev plane; if also $j \geq k,{{(^i}\mathfrak {A})_j}$ and ${{(^{i - k}}\mathfrak {A})_{j - k}}$ are isomorphic. Then if $\mathfrak {A}(i)$ denotes ${{(^i}\mathfrak {A})_{i - 1}},\mathfrak {A}(1), \cdots ,\mathfrak {A}(n)$ is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if ${T_1}, \cdots ,{T_k}$ is an arbitrary sequence of translation planes with common order and if $n \geq k$, then there exists a strongly n-uniform translation Hjelmslev plane $\mathfrak {A}$ of width k such that $\mathfrak {A}(i) \cong {T_i}$ for $i \leq k$. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the $(n - 1)$-uniform planes by means of group congruences.