zbMATH — the first resource for mathematics

Generalizations of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices. (English) Zbl 0939.03075
The well-known one-to-one correspondence between Boolean algebras and Boolean rings can be extended in various ways to a correpondence $$f$$ between the class of orthomodular lattices and certain classes of ring-like structures with two binary operations + and $$\cdot$$ . If one requires that the operations + and $$\cdot$$ can be expressed as terms in the operations of the corresponding orthomodular lattice such that + and $$\cdot$$ coincide with the symmetric difference and meet-operation, respectively, in the case of Boolean algebras and if + and $$\cdot$$ are assumed to be commutative, then there are two “natural” possibilities for choosing + and two “natural” possibilities for $$\cdot$$ , so that all together four mappings arise. It is shown that these mappings are bijections and that the classes of ring-like structures which are assigned to orthomodular lattices by these mappings are varieties. Two of these varieties are explicitly described by the laws defining the variety. Moreover, results are presented concerning implications of the associativity of the operations + and $$\cdot$$ and distributivity of $$\cdot$$ with respect to + to properties of the lattices which correspond to the ring-like structures.

MSC:
 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets 06E20 Ring-theoretic properties of Boolean algebras 08B99 Varieties