Semi-Boolean Algebras Empirical Logic and Rings

摘要

This paper presents application of semi-Boolean algebras to empirical logic and ring theory. The development of semi-Boolean algebras from subtraction algebras is shown and the identity of the two is established. Examples of subtraction algebras are given. A weakening of one of the subtraction axioms leads to a structure which is non-distributive but orthomodular. Known as orthosubtraction algebra, this structure is identical to a semi-orthomodular lattice. Since the subspaces of a Hilbert space (and thus the projections) form an orthomodular lattice they also form an orthosubtraction algebra. Examples of orthosubtraction algebra applied to Hilbert space are given. The concept of a manual and how it relates to empirical logic is introduced next. The set of events of a manual is a semi-Boolean algebra. It is atomic and dominated and has relations of operational complementation and operational perspectivity defined on it. From these relations the manual condition is defined and the semi-Boolean algebra is a DASBAM. Examples of manuals and DASBAMs are given. (Author)