Conditions for the modularity of an orthomodular lattice
David J. Foulis · Pacific Journal of Mathematics · 1961
Introduction.An orthomodular lattice is a lattice L with 0 and 1 which is equipped with an orthocomplementation ': L -• L and which satisfies the orthomodular identity e ^ /=Φ/ = β V (/ Λ e') Recall that an orthocomplementation ': L -> L maps each element e e L onto a complement e' of e in L in such a way that e" = e and e ^/φ /' ^ e' for β,/e L. The ''logic'' of (non-relativistic) quantum mechanics, i.e., the lattice of closed subspaces of a separable infinite dimensional Hubert space [5, p. 49], as well as the "logic" of classical mechanics, i.e., the Boolean algebra of all Borel subsets of phase space modulo Borel subsets of measure zero [5, p. 48], are both instances of orthomodular lattices.L. H. Loomis has shown in [4] that orthomodular lattices provide a natural environment for the abstract study of the dimension theory of operator algebras.I. Kaplansky [3] has obtained an elegant theorem to the effect that if an orthomodular lattice is complete and modular, then it is a continuous geometry.An involution semigroup is a semigroup S equipped with an involution *, i.e., an antiautomorphism * : S-* S oί period 2.An element e e S is called a projection in case e -e % -e 2 .In this paper, we use the term Baer *-semigroup to refer to an involution semigroup S (with a two-sided zero element 0) which is equipped with a mapping ' : S-> S such that (i) x' is a projection for xeS and (ii) for x e S, {y\yes and xy = 0} = x'S.A projection e e S is said to be closed in case e = e", and the collection of all closed projections in S is denoted by P' = P'(S).The notion of a Baer ^-semigroup was introduced in [2, § 2] in a slightly more general form.In [2] it is shown that there is an intimate connection between orthomodular lattices and Baer ^-semigroups, namely: If S is a Baer ^-semigroup, then P'(S) is an orthomodular lattice with e -* e' as orthocomplementation and with partial order defined by e ^ / ef = e for e> feP'(S).The element 0' = 1 acts as a unit in the semigroup S. Conversely, every orthomodular lattice L is isomorphic to a lattice P'(S) for some Bear ^-semigroup S.In the sequel, the symbol L always denotes an orthomodular lattice and the symbol S always denotes a Baer *-semigroup.When S and L are so related that there is an orthocomplementation preserving iso-