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On connections among some orthomodular structures. (English) Zbl 0947.06004
From the introduction: By an orthomodular structure we mean a poset with (partially) defined operations $$\ominus,\oplus$$ such that $$a\leq b\Rightarrow b=a\oplus(b\ominus a)$$, which can be considered as a generalized form of orthomodular law. We study connections between the following orthomodular structures.
Difference posets (effect algebras, alternatively), have been found a useful tool of pursuing quantum mechanical constructions. An important example of a $$D$$-poset is the set of all effects (i.e., s.a. operators $$A$$ with $$0\leq A\leq I$$ on a Hilbert space), which play an important role in unsharp quantum measurements.
The notion of a commutative minimal clan as a common abstraction of Boolean rings and commutative lattice-ordered groups was introduced by Schmidt.
The notion of an MV-algebra was introduced by Chang in 1957, where MV is supposed to suggest many-valued logic. According to Abbott, a meet semi-Boolean algebra is a meet semilattice in which every principal ideal is a Boolean algebra. Abbott has shown that meet semi-Boolean algebras are equationally defined as subtraction algebras.
A close relation between effect algebras and semi-MV-algebras is shown.

##### MSC:
 06C15 Complemented lattices, orthocomplemented lattices and posets 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)