In this paper we introduce the notion of composition hyperring. We show that the composition structure of a composition hyperring is determined by a class of its strong multiendomorphisms. Finally, the three isomorphism theorems of ring theory are derived in the context of composition hyperrings.
Since in a near-ring the distributivity holds just on one side (left or right), it seems naturally to study the behaviour and properties of the set of elements that "correct" the lack of distributivity, in other words that elements that assure the validity of the distributivity. The normal subgroup of the additive structure of a near-ring generated by these elements is called a defect of distributivity of the near-ring. The purpose of this note is to initiate the study of the hypernear-rings (generalizations of near-rings, having the additive part a quasicanonical hypergroup) with a defect of distributivity, making a comparison with similar properties known for near-rings.
In this paper we extend one of the main problems of near-rings to the framework of algebraic hypercompositional structures. This problem states that every near-ring is isomorphic with a near-ring of the transformations of a group. First we endow the set of all multitransformations of a hypergroup (not necessarily abelian) with a general hypernear-ring structure, called the multitransformation general hypernear-ring associated with a hypergroup. Then we show that any hypernear-ring can be weakly embedded into a multitransformation general hypernear-ring, generalizing the similar classical theorem on near-rings. Several properties of hypernear-rings related with this property are discussed and illustrated also by examples.
The notion of (m, n)-ary hyperring was introduced by Davvaz at the 10th AHA
congress [9], as the strong distributive structure. In this article we
generalize it, by introducing the notion of (m, n)-ary hyperring with
inclusive distributivity. We present construction of (m, n)-ary hyperrings
associated with binary relations on semigroup. We also state the condition
under which there exists (m, n)-ary hyperring of multiendomorphisms for a
starting m-ary hypergroup (H, f ). Finaly, we analyze connections between the
obtained classes of (m, n)-ary hyperrings.
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