Giri and Wazalwar evolved concepts of prime ideal and prime radical in noncommutative semigroups. A hemiring is a ring without subtraction (additive inverse), may not have commutativity and identity. A hemiring with identity is called a semiring. It is well known that a hemiring can be embedded in a semiring. We will use this fact to develop proofs of some results on prime radical in a hemiring...

Eilenberg’s variety theorem gives a bijective correspondence between varieties of languages and varieties of finite semigroups. The second author gave a similar relation between conjunctive varieties of languages and varieties of semiring homomorphisms. In this paper, we add a third component to this result by considering varieties of meet automata. We consider three significant classes of lang...

The matrix differential Riccati equation (DRE) is ubiquitous in control and systems theory. The presence of the quadratic term implies that a simple linear-systems fundamental solution does not exist. Of course it is well-known that the Bernoulli substitution may be applied to obtain a linear system of doubled size. Here however, tools from max-plus analysis and semiconvex duality are brought t...

Let S be a semiring and let Z(S)∗ be its set of nonzero zero divisors. We denote the zero divisor graph of S by Γ(S) whose vertex set is Z(S)∗ and there is an edge between the vertices x and y (x 6= y) in Γ(S) if and only if either xy = 0 or yx = 0. In this paper we study the zero divisor graph of the semiring of matrices Mn(B), (n > 1) over the Boolean semiring B. We investigate the properties...

Weighted automata over the tropical semiring Zmax = (Z ∪ {−∞},max,+) are closely related to finitely generated semigroups of matrices over Zmax. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is ...

In the modelling of timed discrete event systems, one traditionally uses dater functions, which give completion times, as a function of numbers of events. Dater functions are non-decreasing. We extend this modelling to the case of multiform logical and physical times, which are needed to model concurrent behaviors. We represent event sequences and time instants by words. A dater is a map, which...

Idempotent mathematics is based on replacing the usual arithmetic operations with a new set of basic operations, that is on replacing numerical fields by idempotent semirings. Exotic semirings such as the max-plus algebra Rmax or concatenation semiring P(Σ) have been introduced in connection with various fields: graph theory, Markov decision processes, language theory, discrete event systems th...

We study the problem of defining normal forms of terms for the algebraic λ-calculus, an extension of the pure λ-calculus where linear combinations of terms are first-class entities: the set of terms is enriched with a structure of vector space, or module, over a fixed semiring. Towards a solution to the problem, we propose a variant of the original reduction notion of terms which avoids annoyin...

Distributed trust management supports the provision of the required levels in a flexible and scalable manner by locally discriminating between the entities with which a principal should interact. However, there is a tension between the preservation of privacy and the controlled release of information when an entity submits credentials for establishing and verifying trust metric where it may dis...

We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. This gives us a “Boolean” cat...