Abstract
On a particular class of m-idempotent hyperrings, the relation ξ m * is the smallest strongly regular equivalence such that the related quotient ring is commutative. Thus, on such hyperrings, ξ m * is a new representation for the α * -relation. In this paper, the ξ m -parts on hyperrings are defined and compared with complete parts, α -parts, and m-complete parts, as generalizations of complete parts in hyperrings. It is also shown how the ξ m -parts help us to study the transitivity property of the ξ m -relation. Finally, ξ m -complete hyperrings are introduced and studied, stressing on the fact that they can be characterized by ξ m -parts. The symmetry plays a fundamental role in this study, since the protagonist is an equivalence relation, defined using also the symmetrical group of permutations of order n.
Highlights
A congruence relation on an algebraic structure is an equivalence relation that is compatible with the given structure, that is, all operations of the structure are well-defined on the equivalence classes
We present several examples that illustrate the fact that ξ m -complete hyperrings are different from ε m -complete hyperrings and αn -complete hyperrings
Ten years after the introduction of the fundamental relation α∗ in [7] on general hyperrings, Norouzi and Cristea [11] defined a new class of m-idempotent hyperrings satisfying a certain condition, where α∗ is no longer the smallest strongly regular relation such that the associated quotient structure
Summary
A congruence relation on an algebraic structure is an equivalence relation that is compatible with the given structure, that is, all operations of the structure are well-defined on the equivalence classes. In 2017, Norouzi and Cristea [11] introduced a particular class of hyperrings where the fundamental relation Γ∗ is not anymore the smallest equivalence such that the associated quotient structure is a ring On this type of hyperrings they defined the fundamental relation ε∗m , smaller than Γ∗ , but with the associated quotient structure non-commutative in general. Already in 1970 Koskas [6] had studied the transitivity property of the β-relation on hypergroups by using the complete parts, that were used as open subsets of suitable topologies on hypergroups They play an important role in defining topological hypercompositional structures [13]. We introduce the class of ξ m -complete hyperrings, characterize them using the ξ m -parts and present their connections with complete, α-complete, and ε-complete hyperrings
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