When does a generalized Boolean quasiring become a Boolean ring?

Generalized Boolean quasirings are ring-like structures used as algebraic models in the foundations of axiomatic quantum mechanics. The quantum mechanical system corresponding to such a quasiring turns out to be a classical one if and only if this quasiring is a Boolean ring with unit. We characterize this situation by a single identity.

It is known that for classical physics the corresponding logical structure are Boolean algebras which are in one-toone correspondence to Boolean rings with unit. Hence it is a natural question to find the difference between Boolean rings with unit and the mentioned ring-like structures. In the following we present an identity which has the property that Recall that a Boolean ring with unit is a unitary ring R = (R, +, ·, 0, 1) that is idempotent, i.e. the identity x x ≈ x holds. (Here and in the following identities are written by using the symbol "≈" which means that the corresponding identity holds for arbitrary elements of the considered algebra(s).) It is well-known that every Boolean ring is commutative and of characteristic 2, i.e. the identity x + x ≈ 0 holds. The notion of a generalized Boolean quasiring was introduced in Dorninger et al. (1997a) as follows: Definition 1 A generalized Boolean quasiring is an algebra R = (R, +, ·, 0, 1) of type (2, 2, 0, 0) satisfying the following identities: The algebra R is called uniquely representable (cf. Dorninger 1998) if it satisfies the identity and a Boolean quasiring (cf. Dorninger et al. 1997a) if it satisfies identity (9) as well as the following identity: x + x ≈ 0.
Remark 2 Identity (1) follows from identities (4) and (9). This means that commutativity of addition follows from unique representability of the corresponding generalized Boolean quasiring.
In Dorninger et al. (1997a) the following result was proved: Theorem 3 A Boolean quasiring R = (R, +, ·, 0, 1) is a Boolean ring with unit if and only if it satisfies the identity Recall that a bounded lattice with an antitone involution is an algebra L = (L, ∨, ∧, , 0, 1) such that (L, ∨, ∧, 0, 1) is a bounded lattice and the following identities are satisfied: The following well-known result (cf. Dorninger et al. 1997a) shows that every uniquely representable generalized Boolean quasiring can be assigned to a bounded lattice with an antitone involution and conversely.
The aim of the present paper is to generalize Theorem 3 for generalized Boolean quasirings as follows: Theorem 5 A uniquely representable generalized Boolean quasiring R = (R, +, ·, 0, 1) is a Boolean ring with unit if and only if it satisfies the identity Remark 6 Identities (3), (5) and (11) together imply (12). This means that Theorem 5 is in fact stronger than Theorem 3.
With the previous results in hand, we obtain now a simple proof of Theorem 5.

Proof of Theorem 5
Let R = (R, +, ·, 0, 1) be a uniquely representable generalized Boolean quasiring satisfying identity (12) and L = (R, ∨, ∧, , 0, 1) denote the corresponding bounded lattice with an antitone involution. According to Lemma 7, L satisfies the identity x ∧ x ≈ 0 and, according to Lemma 8, L is distributive. Hence L is a Boolean algebra and therefore R a Boolean ring with unit. The rest of the proof is clear.