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Let F be any division subring of the real quatei nions ET. Let 12(F) denote the linear space of all square summable sequences from F and let L denote the lattice of all a I-closed" subspaces of 12(F), where I" denotes the orthogonality relation derived from the H-valued form (a, b) = i aji6 where a, bel2(F), a=(as; i=1, 2, * * ) and b=(bi; i=1, 2, * * * ). Then L is complete, orthocomplemented, M-symmetric, irreducible, atomistic, and separable, but L is orthomodular if and only if F is either the reals, the complex numbers, or the quaternions. The lattice of all closed subspaces of infinite-dimensional, separable, complex Hilbert space has these seven lattice-theoretic properties: (i) complete [1, p. 6]; (ii) orthocomplemented [1, p. 52], [2, p. 42]; (iii) atomistic (Every element is the join of the atoms beneath it.) [3, p. 48]; (iv) irreducible (The center consists precisely of 0 and 1.) [1, p. 67], [3, p. 27]; (v) separable (An orthogonal family of atoms has at most countably many elements.) and infinite dimensional [3, p. 58 ]; (vi) M-symmetric (If a, b EL, we write aMb if x<b implies xV(aAb) =(xVa)Ab. L is M-symmetric if aMb implies bMa.) [1, p. 82], [3, p. 2]; (vii) orthomodular (If a, bEL and a <b, then b=aV(bAa').) [1, p. 53], [2, p. 42]. Real and quaternionic Hilbert space have the same properties. The question arises whether these are the only three lattices (up to ortho-isomorphism) having them. The problem underlying this question is one of coordinatization, that is, the realization of an abstract lattice, described only by algebraic properties, as a lattice associated in some natural way with a concrete object, for example, the lattice of projections of Hilbert space. Work towards a coordinatization theorem for lattices with the above properties has been done Received by the editors May 4, 1970. AMS 1969 subject classifications. Primary 0640; Secondary 8106.

The orthomodular identity and metric completeness of the coordinatizing division ring