## The structure of $n$-uniform translation Hjelmslev planes

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- by David A. Drake PDF
- 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 2

*xk*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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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**175**(1973), 249-282 - MSC: Primary 50D30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0310755-0
- MathSciNet review: 0310755