Abstract
In this paper we introduce the notion of generalized T-semiaffine linear space of finite dimension at least three, T being a suitable set of non–negative integers, and discuss generalized [s, t]-semiaffine linear spaces for suitable \(s \le t\). We will present theorems on generalized \(\{0, 1\}\)-semiaffine linear spaces whose lines have length at least 4 and on finite generalized [0, 2]-semiaffine linear spaces, improving known results of Van Maldeghem and Kreuzer. In particular, finite generalized [0, 2]-semiaffine linear spaces whose lines have at least nine points are classified.
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Notes
The sum of two linear spaces \(\mathbb {L}=(\mathcal P, \mathcal L)\) and \(\mathbb {L}'=(\mathcal P', \mathcal L')\) is the linear space \(\mathbb {L}\oplus \mathbb {L}'\) whose points are those of \(\mathcal P\) and \(\mathcal P'\) and whose lines are the elements of \(\mathcal L \cup \mathcal L'\) and all the 2-sets \(\{x, y\}\), with \(x \in \mathcal P\) and \(y \in \mathcal P'\). A linear space is said to be degenerate if, and only if, it is the sum of two linear spaces.
If s and t are two fixed non-negative integers such that \(s \le t\), then [s, t] will denote the set of all integers i such that \(s \le i \le t\).
The classification of Lo Re and Olanda provides exceptional cases of [0, 2]-semiaffine planes, but these planes contain lines of length three.
The classification of Ohler and Pickert provides just one exceptional case of \(\{1,2\}\)-semiaffine plane, the Shrikhanda plane, but this plane contains lines of length three.
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Ferrara-Dentice, E., Iannotta, G. Generalized Semiaffine linear spaces. Ricerche mat 66, 395–406 (2017). https://doi.org/10.1007/s11587-016-0306-8
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DOI: https://doi.org/10.1007/s11587-016-0306-8