A vector lattice description of a simple quantum system. Pt.1

The cannonical 8-dimensional discrete vector lattice (A, EPSILON, E) with a positive cone EPSILON and an order unit E is considered as the space of random variables over the Boolean algebra 28. The ordered triple of basic variables is determined using the properties of a free Boolean algebra with three free generating elements. It is shown that all triples of basic variables in (A, EPSILON, E) are obtained from one another through the symmetries of the cone as well as that each orthonormal basis generated by basic variables and their products corresponds to a basis of eight atoms through an orthogonal and involuntary transformation. The orthogonal representation of the three-dimensional rotation group in A is considered and the correspondence between the triple of basic variables and the right-handed Cartesian space coordinate systems is referred to as the transformation law for the basic variables. It is shown that such an orthogonal representation of O(3,R) acts on both the lattice cone and the basic variables. It is claimed that the corresponding physical system is the quantum one known as spin-1/2. The classical (and therefore commutative) description is not given. The non-commutativity of the quantum observables is replaced by a property which distinguishes the model both from classical and quantum mechanics. The model is an extention of standard formalism and is reduced to the operator description