On connections among some orthomodular structures.

*(English)*Zbl 0947.06004From 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.

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) |