Lattice properties of subspace families in an inner product space

Let S be a separable inner product space over the field of real numbers. Le t E(S) (resp., C(S)) denote the orthomodular poset of all splitting subspac es (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 Hilb ert). This question is relevant to the recent study of the algebraic proper ties of splitting subspaces and to the search for "nonstandard" orthomodula r 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 s howing 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) wi th E(S) being a (modular) lattice. (Thus, the lattice properties of E(S) (r esp. 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-produ ct 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((S) over bar) for (S) ov er bar being the completion of S (this provides a solution to a recently fo rmulated problem).