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Transactions of the American Mathematical Society

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



The structure of $ n$-uniform translation Hjelmslev planes

Author: David A. Drake
Journal: Trans. Amer. Math. Soc. 175 (1973), 249-282
MSC: Primary 50D30
MathSciNet review: 0310755
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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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Keywords: Translation, translation plane, group congruence, Hjelmslev plane, n-uniform, strongly n-uniform, height n, level n
Article copyright: © Copyright 1973 American Mathematical Society