When does a generalized Boolean quasiring become a Boolean ring? (Q1626224)

Generalized Boolean quasirings (GBQRs) are ringlike structures \(\langle G; \cdot, +, 0, 1\rangle\), which were introduced to develop axiomatic quantum mechanics. A GBQR is called uniquely representable if it satisfies the identity \((1 + (1 + x)(1 + y))(1 + x y) \approx x + y\). The usual correspondence between Boolean rings and Boolean algebras lifts to a correspondence between uniquely representable GBQRs and bounded lattices with antitone involution. The former may be viewed as the distributive version of the latter. In fact, it is known that \(x(1+y)\approx x+xy\) is enough distributivity to guarantee that a uniquely representable GBQR is a Boolean ring. In this paper, the authors prove that a uniquely representable GBQR satisfying a still weaker fragment of distributivity, \(x(1+xy)\approx x+xy\), is a Boolean ring.