Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationa...

We propose a new class of mathematical structures called (m, n)-semirings (which generalize the usual semirings) and describe their basic properties. We define partial ordering and generalize the concepts of congruence, homomorphism, and so forth, for (m, n)-semirings. Following earlier work by Rao (2008), we consider systems made up of several components whose failures may cause them to fail a...

We study quasi completely regular semirings through their congruence structures.

We provide the operations of matrix addition, multiplication, transposition, and matrix comparisons as executable functions over ordered semirings. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over matrices. We further show that the standard semirings over the naturals, integers, and rationals, as well as the arctic s...

Systems of polynomials on semirings arise in several branches of computer science, like static analysis of procedural programs or formal language theory. We propose a new technique for calculating the least fixed points of such polynomial systems. This technique is a generalization of Newton’s method, the well-known method for approximating a zero of a nonlinear function on the reals. We show t...

One of the most well-known induction principles in computer science is the fixed point induction rule, or least pre-fixed point rule. Inductive ∗semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. Inductive ∗-semirings are extensions of continuous semirings and the Kleene algebras of Co...