Lattice properties of subspace families in an inner product space

Let $S$ be a separable inner product space over the field of real numbers. Let $E(S)$ (resp., $C(S))$ denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of $S$. We ask whether $E(S)$ (resp., $C(S))$ can be a lattice without $S$ being complete (i.e. without $S$ being Hilbert). This question is relevant to the recent study of the algebraic properties of splitting subspaces and to the search for ``nonstandard'' orthomodular spaces as motivated by quantum theories. We first exhibit such a space $S$ that $E(S)$ is not a lattice and $C(S)$ is a (modular) lattice. We then go on showing that the orthomodular poset $E(S)$ may not be a lattice even if $E(S)=C(S)$. Finally, we construct a noncomplete space $S$ such that $E(S)=C(S)$ with $E(S)$ being a (modular) lattice. (Thus, the lattice properties of $E(S)$ (resp. $C(S))$ do not seem to have an explicit relation to the completeness of $S$ though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete $S$ such that all states on $E(S)$ are restrictions of the states on $E(\overline{S})$ for $\overline{S}$ being the completion of $S$ (this provides a solution to a recently formulated problem).