66 research outputs found
On the relation between hyperrings and fuzzy rings
We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields
A Brief Survey on the two Different Approaches of Fundamental Equivalence Relations on Hyperstructures
Fundamental structures are the main tools in the study of hyperstructures. Fundamental equivalence relations link hyperstructure theory to the theory of corresponding classical structures. They also introduce new hyperstructure classes.The present paper is a brief reference to the two different approaches to the notion of the fundamental relation in hyperstructures. The first one belongs to Koskas, who introduced the β ∗ - relation in hyperstructures and the second approach to Vougiouklis, who gave the name fundamental to the resulting quotient sets. The two approaches, the necessary definitions and the theorems for the introduction of the fundamental equivalence relation in hyperstructures, are presented
absorbing prime hyperideals in multiplicative hyperrings
In this paper, we define the concept prime hyperideal in a multiplicative
hyperring . A proper hyperideal of is an prime hyperideal if for
with implies or . We
provide some characterizations of prime hyperideals. Also we conceptualize
and study the notions absorbing prime and absorbing prime
hyperideals into multiplicative hyperrings as generalizations of prime ideals.
A proper hyperideal of a hyperring is an absorbing prime
hyperideal if for such that , then
for some . We study some properties of such
generalizations. We prove that if is an prime hyperideal of a hyperring
, then each of , , , ,
and are prime hyperideals under suitable conditions and suitable
hyperideal , where is a hyperideal contains in . Also, we
characterize prime hyperideals in the decomposite hyperrings. Moreover, we
show that the hyperring with finite number of maximal hyperideals in which
every proper hyperideal is absorbing prime is a finite product of
hyperfields.Comment: Journal of algebraic system
History and new possible research directions of hyperstructures
We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research
Hopf algebras for matroids over hyperfields
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields
Semiring systems arising from hyperrings
Hyperfields and systems are two algebraic frameworks which have been
developed to provide a unified approach to classical and tropical structures.
All hyperfields, and more generally hyperrings, can be represented by systems.
We show that, conversely, we show that the systems arising in this way, called
hypersystems, are characterized by certain elimination axioms. Systems are
preserved under standard algebraic constructions; for instance matrices and
polynomials over hypersystems are systems, but not hypersystems. We illustrate
these results by discussing several examples of systems and hyperfields, and
constructions like matroids over systems.Comment: 29 p
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