For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

This addendum to [2] shows that the set of tautological quantum logical propositional formulas for a finite dimensional vector space Cⁿ is different for every n, affirmatively answering a question posed therein.

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame of the (...) logic and therefore characterises orthomodular quantum logic. To prove soundness, a further restriction is placed on admissible valuations which requires that they be point-generated in addition to the requirement that they return stable sets. (shrink)