the concept of  -semihyperring is a generalization of semiring, a generalization of semihyperring and a generalization of      -semiring. since the theory of ideals plays an important role in the theory of  - semihyperring, in this paper, we will make an intensive study of the notions of noetherian, artinian, simple and   regular      -semihyperrings. the bulk of this paper...

1 Abstract We show that the equatorial semiring Z min = (Zf+1g; min; +) is a Fatou extension of the tropical semiring M = (Nf+1g; min; +). This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring. We also deduce from it the decidability of the limitedness problem for the equatorial semiring, solving th...

A semiring can be “completed” (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is “natural” (a left adjoint to the forgetful functor), and quite straightforward. Definitions and easy facts Definition 1. A semiring is a structure (S,+, ·, 0, 1) such that (S,+, 0) ...

Schmets [22] has developed a measure theory from a generalized notion of a semiring of sets. Goguadze [15] has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota [20] has defined this quasi-semiring. We propose ...

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...

Let $R$ be a prime ring, $I$ be any nonzero ideal of $R$ and $f:Irightarrow R$ be an additivemap. Then skew-Engel condition $langle... langle langle$$f(x),x^{n_1} rangle,x^{n_2} rangle ,...,x^{n_k} rangle=0$ implies that $f (x)=0$ $forall,xin I$ provided $2neq$ char $(R)>n_1+n_2+...+n_k, $ where $n_1,n_2,...,n_k$ are natural numbers. This extends some existing results. In the end, we also gener...

For several semirings S, two weighted finite automata with multiplicities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called “proper”. It is known that the Boolean semiring, the semiring of natural numbers, the ring of integers, all finite commutative positively ordered semirings and all fields are proper. The semiring S is Noetherian...