A characterization of Grassmann and attenuated spaces as \((0,\alpha)\)-geometries. (Q1399685)

The authors consider \((0,\alpha)\)-geometries with special properties in order to obtain a characterization of Grassmann and attenuated spaces. A finite \((0,\alpha)\)-geometry (\(\alpha\geq 1\), integer) with parameters \((s,t)\) is a connected semilinear space satisfying the following conditions: Each block contains \(s+1\) points; each point belongs to \(t+1\) blocks; for every antiflag \((x,B)\), \(\alpha(x,B)\) is equal to 0 or \(\alpha\). (Here \(\alpha(x,B)\) denotes the number of points on \(B\) adjacent to \(x\).) If a finite \((0,\alpha)\)-geometry satisfies a certain regularity condition (with parameter \(\mu> 0\)) then it is called an amply regular \((0,\alpha)\)-geometry with parameters \(s\), \(t\), \(\alpha\), \(\mu\). The main result is a common characterization of the Grassmann and the attenuated spaces as ample regular \((0,\alpha)\)-geometries satisfying the dual Veblen-Young axiom \((DV^*)\) and certain conditions referring to \(s\), \(t\), \(\alpha\), \(\mu\).