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Let R be a commutative semiring with identity and M be a unitary Rsemimodule. Let φ : S(M) → S(M) ∪ {∅} be a function, where S(M) is the set of all subsemimodules of M. We say a proper subsemimodule N of M is φ-primary subsemimodule, if r ∈ R, x ∈ M and rx ∈ N − φ(N) imply that r ∈ √(N :R M) or x ∈ N. The notion of φ-primary subsemimodules is a generalization of the concept of primary, weakly primary and φ-prime subsemimodules. We study properties of φ-primary subsemimodules of a semimodule M and related results to those of ring theory. [ABSTRACT FROM AUTHOR]

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Let S be a commutative semiring with unity different than zero. In this paper, we study the concept of 2-absorbing ideals of S which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a 2-absorbing ideal is also a 2-absorbing ideal and there are at most 2 prime k-ideals of S that are minimal over a 2-absorbing ideals. [ABSTRACT FROM AUTHOR]

We give a lower bound on the speed at which Newton's method (as defined in [11] ) converges over arbitrary ω -continuous commutative semirings. From this result, we deduce that Newton's method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ N ” (i.e. k = k + 1 holds) in the sense of Bloom and Ésik [2] . We apply these results to (1) obtain a generalization of Parikh's theorem, (2) compute the provenance of Datalog queries, and (3) analyze weighted pushdown systems. We further show how to compute Newton's method over any ω -continuous semiring by constructing a grammar unfolding w.r.t. “tree dimension”. We review several concepts equivalent to tree dimension and prove a new relation to pathwidth. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigatebisemialgebrasandHopf semialgebrasover commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over commutative rings including the main reconstruction theorems and theFundamental Theorem of Hopf Algebras. We also provide a notion ofquantum monoidsas Hopf semialgebras which are neither commutative nor cocommutative; this extends the Hopf algebraic notion of a quantum group. The generalization to the semialgebraic context is neither trivial nor straightforward due to the nonadditive nature of the base category of commutative monoids which is also neither Puppe-exact nor homological and does not necessarily have enough injectives. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate semicorings over associative semirings and their categories of semicomodules. Our results generalize old and recent ones on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at several places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no nonzero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings. [ABSTRACT FROM AUTHOR]

Using a restrictive notion of exactness and the natural tensor product, we generalize several results related to flat modules over rings to flat semimodules over semirings. [ABSTRACT FROM AUTHOR]

Abstract: We introduce and study an abstract class of semirings, which we call exact semirings, defined by a Hahn–Banach-type separation property on modules. Our motivation comes from the tropical semiring, and in particular a desire to understand the often surprising extent to which it behaves like a field. The definition of exactness abstracts an elementary property common to both fields and the tropical semiring, which we believe is fundamental to explaining this similarity. The class of exact semirings turns out to include many other important examples of both rings (proper quotients of principal ideal domains, matrix rings and finite group rings over these and over fields), and semirings (the Boolean semiring, generalisations of the tropical semiring, matrix semirings and group semirings over these). [Copyright &y& Elsevier]

Theory of rough sets is an important tool in data mining. In this paper, we are initiating the study of roughness in hemirings with respect to the Pawlak approximation space and also with respect to the generalized approximation space. Lower and upper rough subhemirings and ideals are studied. Lower and upper approximations of k-ideals and h-ideals of a hemiring are discussed. [ABSTRACT FROM AUTHOR]

In [7], we have introduced the concept of exact sequences of R-semimodules and R-homomorphisms. Here we give some interesting generalizations of exact sequences of modules and module homomorphisms over rings. [ABSTRACT FROM AUTHOR]

From an algebraic point of view, semirings provide the most natural generalizations of the theory of rings. In this paper, the ring theoretic results of [8] concerning the primary ideals and their radicals to noncommutative semirings are generalized. The derived results are further carried over to a Gel'fand semiring endowed with finite group action. [ABSTRACT FROM AUTHOR]