Unlike the field of real numbers over multiplication and addition, the set of boolean numbers does not constitute a field over the operations of conjunction and exclusive-or (there is no inverse element over conjunction). What is found however is that, as every boolean algebra gives rise to a boolean ring and vice-versa as seen [here], we can extend the notion of boolean rings to that of boolean modules (B-molules).
As was seen in [CM’s], logical expressions can be treated as a sort of vector which are measured by a correspondence matrix (CM). This paper introduces the notion for b-modules as well as develops a mathematical foundation for the logical state-vectors used to construct CM’s. We give the notion of a ‘Bector’ as an element of a ‘Bector-space’ which is defined over boolean rings. Bectors are analogous to vectors defined over fields and are the true logical state-vectors as discussed previously.
This work has the potential to extend way beyond its intended purpose which was to create boolean like vectors. Modules theory is used extensively in Algebraic Geometry and this papers development of the B-module may possibly bridge the field with propositional and other types of logic.
