In this paper, we initiate the study of prime bi-ideals (fuzzy bi-ideals) in semirings. We define strongly prime, prime, semiprime, irreducible and strongly irreducible bi-ideals in semirings. We also define strongly prime, semiprime, irreducible, strongly irreducible fuzzy bi-ideals of semirings. We characterize those semirings in which each bi-ideal (fuzzy bi-ideal) is prime (strongly prime).

A categorical equivalence between algebraic contexts with relational morphisms and join-semilattices with homomorphisms is presented and extended to idempotent semirings and domain semirings. These contexts are the Kripke structures for idempotent semirings and allow more efficient computations on finite models because they can be logarithmically smaller than the original semiring. Some example...

In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of ne...

2 of Lemma 4 and 4 of Proposition 2 in our paper [1] entitled “*Continuous Idempotent Semirings and Their Ideal Completion” are wrong. This note shows counter examples of them. The counter examples are given in linguistic models of *-Continuous Idempotent Semirings. 1 Monodic Tree Languages A ranked set is a couple (Σ, r) of a set Σ of (function) symbols and a mapping r from Σ to the set N of n...

As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. power-series as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable homomorphism. Recently, we presented the idea of approximating the least solution of such algebraic s...

A *–k-semiring is an ordered semiring S equipped with a star operation such that for any a,b 2 S, a*b is the least fixed point of the linear mapping x # ax + b over S. The notion of *–k-semirings is a generalization of several important concepts such as continuous semirings, (weak) inductive *-semirings and the Kleene algebras of Conway and Kozen. We investigate several basic properties of *–k-...

Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, Löb’s formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence ...

We define rationally additive semirings that are a generalization of (ω)complete and (ω-)continuous semirings. We prove that every rationally additive semiring is an iteration semiring. Moreover, we characterize the semirings of rational power series with coefficients in N∞, the semiring of natural numbers equipped with a top element, as the free rationally additive semirings.

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimizat...