On the n-complete hypergroups

We study existence and possible uniqueness of special semihypergroups of type U on the right. In particular, we prove that there exists a unique proper semihypergroup of this kind having order 6, apart of isomorphisms; the least order for a hypergroup of type U on the right to have a stable part which is not a subhypergroup is 9; and the minimal cardinality of a proper semihypergroup of that kind whose heart and derived semihypergroup are proper and nontrivial is 12.Contextually, we analyze properties of the kernel of homomorphisms g : H → G, where H is a finite semihypergroup of type U on the right and G is a group. In this way, we obtain results that are immediately applicable both to the heart and to the derived of such semihypergroups.

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“…Proof. If H is a hypergroup, the result is already known, see [4], [8], [16]. Then, suppose that H is a proper semihypergroup.…”

Section: Fundamental Relations In Semihypergroups Of Type U On the Rightmentioning

confidence: 97%

“…As a consequence, we obtain β ⊂ γ, and moreover, β * ⊂ γ * . Finally, note that the canonical projections ϕ : H → H/β * and ψ : H → H/γ * are good homomorphisms, and if H is a hypergroup, the kernels of ϕ and ψ are the heart and the derived hypergroup of H respectively, see [4], [8], [9], [16].…”

Section: Fundamental Relations In Semihypergroupsmentioning

confidence: 99%

Minimal Order Semihypergroups of Type U on the Right

Fasino

Freni

2008

MedJM

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“…Certain equivalence relations, called fundamental relations, introduce natural correspondences between algebraic hyperstructures and classical algebraic structures. These equivalence relations have the property of being the smallest strongly regular equivalence relations such that the corresponding quotients are classical algebraic structures [4][5][6][7][8][9][10][11]. For example, if (H, •) is a hypergroup, then the fundamental relation β is transitive [12][13][14] and the quotient set H/β is a group.…”

Section: Introductionmentioning

confidence: 99%

On Hypergroups with a β-Class of Finite Height

et al. 2020

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In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.

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“…One important class of hyperrings was introduced by Rota in 1982, where the multiplication is a hyperoperation, while the addition is an operation, which is called multiplicative hyperrings (for more details see [32,33,34,35]) and was subsequently investigated by Olson and Ward [26] and many others. De Salvo [15] introduced hyperrings in which the additions and the multiplications are hyperoperations. Moreover, there exist other types of hyperrings where both the addition and multiplication are hyperoperations and instead associativity, commutativity and distributivity satisfy weak associativity, weak commutativity and weak distributivity.…”

Section: Introductionmentioning

confidence: 99%