Algebraic aspect of the discrete maximum principle

We discuss the discrete maximum principle in the formal language of linear algebra (not using any concepts of the difference scheme theory). Due to this formal algebraic approach it is not difficult to find the presence or the absence of conditions for the realization of the maximum principle in any algorithm represented as the system of linear algebraic equations. Using the general theorems formulated, it is possible to systematically construct approximate algorithms satisfying the discrete maximum principle, i.e. absolutely stable ones. We also consider the case in which the discrete maximum principle is extended to 'inhomogeneous' algebraic systems that approximate equations with nonzero right-hand side in initial problems. The theoretical results are illustrated by examples of the discretizations for typical problems of mathematical physics.