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It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular lattice  X , admits a natural embedding into a groupG  ⁢   (  X  ){G(X)}    with a lattice ordering so thatG  ⁢   (  X  ){G(X)}    determines  X  up to isomorphism. The embeddingX  ↪   G  ⁢   (  X  ){X\hookrightarrow G(X)}    appears to be a universal (non-commutative) group-valued measure on  X , while states of  X  turn into real-valued group homomorphisms onG  ⁢   (  X  ){G(X)}   . The existence of completions is characterized by a generalized archimedean property which simultaneously applies to  X  andG  ⁢   (  X  ){G(X)}   . By an extension of Foulis’ coordinatization theorem, the negative cone ofG  ⁢   (  X  ){G(X)}    is shown to be the initial object among generalized Baer∗{{}^{\ast}}   -semigroups. For finite  X , the correspondence between  X  andG  ⁢   (  X  ){G(X)}    provides a new class of Garside groups.

Von Neumann algebras, L -algebras, Baer *-monoids, and Garside groups