# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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## The structure of $n$-uniform translation Hjelmslev planesHTML articles powered by AMS MathViewer

by David A. Drake
Trans. Amer. Math. Soc. 175 (1973), 249-282 Request permission

## 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.
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