Orthomodular lattices of subspaces obtained from quadratic forms (Q1768724)

Let \(E\) be a 3-dimensional vector space over some field \(K\) (\(2 \neq \operatorname{char} K\neq 3\)), \(\varphi\) a non singular symmetric bilinear form on \(E\). The authors consider the modular lattice \(L(E,\varphi)\) of subspaces of \(E\), i. e. essentially the projective plane over \(K\), equipped with a polarity. The subset \(T(E,\varphi)\) of non-isotropic subspaces is not a sublattice of \(L(E,\varphi)\) but a lattice with respect to the inclusion order, and \(M\rightarrow M^\bot\) an orthocomplementation on \(T(E,\varphi)\). The authors show: (Thm. 2.2) \(T(E,\varphi)\) is an orthomodularlattice, and modularity of \(T(E,\varphi)\) is equivalent to \(T(E,\varphi)=L(E,\varphi)\). Their main result (Thm. 3.3) concerns the irreducible 3-homogeneous subalgebras of \(T(E,\varphi)\). They prove that if there exist a subfield \(K'\) of \(K\), a \(K'\)-closed orthogonal base of \(E\), a \(K'\)-subspace \(E'\) of \(E\), and a bilinear form \(\varphi'\) induced by \(\varphi\), then \(T(E',\varphi')\) is an irreducible 3-homogeneous subalgebra of \(T(E,\varphi)\). Furthermore they sketch a proof of their claim that any irreducible 3-homogeneous subalgebra \(T'\) of \(T(E,\varphi)\) is isomorphic to some \(T(E', \varphi')\). This theorem generates a finite minimal orthomodular lattice \(L\) (i.e. \(L\) is not modular, but all of its proper subalgebras are either modular, or isomorphic to \(L\)) for any prime \(>4\), and an infinite one using the field of rational numbers. This continues the work on minimal orthomodular lattices begun by \textit{J. C. Carrega, R. J. Greechie} and \textit{R. Mayet} [Int. J. Theor. Phys. 39, No. 3, 517--524 (2000; Zbl 0963.06008)].