In recent years, soft sets have become popular in various fields. For this reason, many studies been carried out the field of algebra. study, intersection k-ideals are defined with help a semiring, and some algebraic structures examined. Moreover, quotient rings by k-semiring. Isomorphism theorems examined rings. Finally, properties investigated defining maximal k-ideals.

Assumed that (S,+,.) is a semiring. Semiring algebra structure as generalization of ring. A set I⊆S called an ideal over semiring S if for any α,β∈I, we have α-β∈I and sα=αs∈I every s in S. Based on this definition, there special condition namely prime P, when αβ∈P, then could prove α or β are elements P. Furthermore, I irreducible Ia intersection from B S, I=A I=B. We also know the strongly no...

Aiming for a systematic feature extraction from time series, we introduce the iterated-sums signature over arbitrary commutative semirings. The case of tropical semiring is central, and our motivating, example. It leads to features (real-valued) series that are not easily available using existing signature-type objects. We demonstrate how extracts chronological aspects its calculation possible ...

A Kleene algebra (K,+, ·,∗ , 0, 1) is an idempotent semiring with an iteration ∗ as axiomatised by Kozen. We consider left semiring modules (A,+, 0, :) over Kleene algebras. We call such a left semiring module a Kleene module if each linear equation x = a + r : x has a least solution, where : is the product from K × A to A. The linear context-free languages can be viewed as a Kleene module A ov...

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial sλ(x1, . . . , xk) labeled by a partition λ = (λ1 ≥ λ2 ≥ · · · ) is bounded by O(log(λ1)) provided the number of variables k is fixed.

Overview Diagnosis traditionally viewed as logical reasoning – (de Kleer and Williams 87), (Reiter 87), ... But more naturally viewed as constraint optimization – Minimal set of faulty components, most likely fault, ... Framework that unifies qualitative and quantitative notions of diagnosis using semiring-based CSP – Choose appropriate semiring and construct constraints Diagnosis algorithms ba...