This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring – a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra....

Abstract. The interval-valued prime fuzzy ideals (in brevity, the iv. prime fuzzy ideals) of a semigroup have been recently studied by Kar, Sarkar and Shum [18]. As a continued study of i-v fuzzy ideals, we are going to investigate the properties of i-v fuzzy quasi-ideal and i-v fuzzy bi-ideal of a semiring and then we characterize the regularity and intra-regularity of a semiring in terms of t...

Spatiotemporal trajectories are coded by "theta sequences," ordered series of hippocampal place cell spikes that reflect the order of behavioral experiences. Theta sequences are thought to be organized by co-occurring gamma rhythms (∼25-100 Hz). However, how sequences of locations are represented during distinct slow (∼25-55 Hz) and fast (∼60-100 Hz) gamma subtypes remains poorly understood. We...

The semiring-based formalism to model soft constraint has been introduced in 1995 by Ugo Montanari and the authors of this paper. The idea was to make constraint programming more flexible and widely applicable. We also wanted to define the extension via a general formalism, so that all its instances could inherit its properties and be easily compared. Since then, much work has been done to stud...

From now on X denotes a set and S denotes a family of subsets of X. Now we state the propositions: (1) Let us consider sets X1, X2, a family S1 of subsets of X1, and a family S2 of subsets of X2. Then {a×b, where a is an element of S1, b is an element of S2 : a ∈ S1 and b ∈ S2} = {s, where s is a subset of X1 ×X2 : there exist sets a, b such that a ∈ S1 and b ∈ S2 and s = a × b}. Proof: {a × b,...

The basic aim of key exchange is that two people who can only communicate via an insecure channel want to find a common secre t key without any attack. In this paper we show a model of key-agreement protocol using polynomials over non-commutative division semiring for network security. It is proved that the proposed protocol meets several security attributes under the assumption that the polyno...

We investigate the theory of skew (formal) power series introduced by Droste, Kuske [4, 5], if the basic semiring is a Conway semiring. This yields Kleene Theorems for skew power series, whose supports contain finite and infinite words. We then develop a theory of convergence in semirings of skew power series based on the discrete convergence. As an application this yields a Kleene Theorem prov...

The problem of computing the edit-distance between a string and a finite automaton arises in a variety of applications in computational biology, text processing, and speech recognition. This paper presents linear-space algorithms for computing the edit-distance between a string and an arbitrary weighted automaton over the tropical semiring, or an unambiguous weighted automaton over an arbitrary...

We investigate modal algebras that arise by adding operations of domain and antidomain to semirings and Kleene algebras. Let (S,+, ·, 0, 1) be a noncommutative and additively idempotent semiring with natural order defined as x ≤ y ⇔ x + y = x. Such semirings have many computational models, e.g., binary relations under union and relative product, languages, paths in graphs or program traces unde...