Comparison and classification of linear systems over idempotent semirings inspired by total positivity

To the best knowledge of the author, this paper is the first attempt to develop the theory of total positivity for linear dynamical systems over idempotent semirings which will be denoted ITP. More precisely, in this paper we study the analog of total positivity of order 2 concept for matrices which entries are in an idempotent semiring denoted by ITP2. The idempotent version of the basic composition formula of Polya and Szegö in the particular case of ITP2 matrices is proved in this paper. From this main result we show that the ITP2 concept plays a central role in order to classify elements of idempotent semimodules and monotonicity of linear systems over idempotent semirings which allows their comparisons. This paper has mainly benefited from the guidelines of reliability theory and statistical tests hypothesis theory. Finally, let us mention that in the context of combinatorial optimization or recognition problems the total positivity of order 2 property is known as the Monge property. This remark could lead to further work in different topics of research.