Term extensions of partial ring-like quantum logics

Abstract

The one-to-one correspondence between Boolean algebras and Boolean rings can be generalized to a mapping from partial ring-like algebras to bounded lattices with an antitone involution ( * -lattices), cf.[1], [2]. These partial algebras are known as partial generalized Boolean quasirings (pGBQRs) and can be used to construct various kinds of quantum logics (cf.[1] -[6]). To a large extent this is due to the fact that the partial operation ⊕ of pGBQRs can be extended almost arbitrarily to a full operation +. However, in view of Boolean rings and Boolean algebras it is evident to demand that + should be a term built up by the operations of the corresponding * -lattice. How this can be achieved is investigated as well for the case that the underlying lattice is an arbitrary * -lattice as for the special case of de Morgan algebras. Moreover, we study various classes of pGBQRs such that + is associative.Mathematics Subject Classification: 08A40, 06C15, 06D30