Augmented polynomial matrices and algebraization of switching circuits

Over rings of polynomials with idempotent variables (over arbitrary fields) there are defined classes of augmented matrices (with one distinguished column) that realize Boolean functions. In the latter classes of augmented matrices (over any fields) there is defined a system of equivalent transformations (preserving realized Boolean functions) that generalizes the known system of elementary transformations (of rows and columns) of usual polynomial matrices. It is proved the completeness of this system for the simplest (binary) case – in the class of augmented matrices over the ring of Zhegalkin polynomials. In particular, there is given a method for reducing of an arbitrary augmented matrix over the ring of Zhegalkin polynomials by means of this system to a uniquely determined one-element form. For the same (binary) case, it is shown that the class of binary incidence matrixes of switching circuits is, in essence, a subclass of the class of augmented matrices over the ring of Zhegalkin polynomials. This reveals the simplest «completely algebraic» extension of the class of switching circuits – one of the basic model classes of mathematical theory of control systems.