On hypercyclic fully zero-simple semihypergroups

Salvo, Mario De; Freni, Domenico; Faro, Giovanni Lo

doi:10.3906/mat-1904-14

Cited by 6 publications

(6 citation statements)

References 14 publications

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“…The relations β, β * are called fundamental relations on H [1,9,13,14,18]. The interested reader can find all relevant definitions, many properties and applications of fundamental relations, even in more abstract contexts, also in [4,5,7,8,10,15].…”

Section: Basic Definitions and Resultsmentioning

confidence: 99%

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Semihypergroups obtained by merging of 0-semigroups with groups

Mario¹,

Fasino²,

Freni³

et al. 2018

Filomat

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We consider the class of 0-semigroups (H,) that are obtained by adding a zero element to a group (G, •) so that for all x, y ∈ G it holds x y 0 ⇒ x y = xy. These semigroups are called 0-extensions of (G, •). We introduce a merging operation that constructs a 0-semihypergroup from a 0-extension of (G, •) by a suitable superposition of the product tables. We characterize a class of 0-simple semihypergroups that are merging of a 0-extension of an elementary Abelian 2-group. Moreover, we prove that in the finite case all such 0-semihypergroups can be obtained from a special construction where (H,) is nilpotent.

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Section: Basic Definitions and Resultsmentioning

confidence: 99%

“…By definition(6) and(8), for any x, y ∈ H we have x * y = x y if x y or x ∈ {0, 1} or y ∈ {0, 1}. Moreover, if x ∈ H H − {0, 1} then there exist a, b {0, 1} such that a x b and x = a b = ab.…”

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confidence: 99%

Semihypergroups obtained by merging of 0-semigroups with groups

Mario¹,

Fasino²,

Freni³

et al. 2018

Filomat

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“…Analogous is the variety of the substructures which are revealed here in the case of the left/right almost-hypergroups. For the consistency of the terminology [4,13,33,55,[91][92][93][94][95][96][97][98][99], the terms semisub-left/right almost-hypergroup, sub-left/right almost-hypergroup, etc. will be used in exactly the same way as the prefixes sub-and semisub-are used in the cases of the groups and the hypergroups, e.g., the terms subgroup, subhypergroup are used instead of hypersubgroup, etc.…”

Section: Substructures Of the Left/right Almost-hypergroupsmentioning

confidence: 99%

On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations

Massouros

Yaqoob

2021

Mathematics

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This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortified transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being.

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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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On hypercyclic fully zero-simple semihypergroups

Abstract: Let I be the class of fully zero-simple semihypergroups generated by a hyperproduct. In this paper we study some properties of residual semihypergroup (H+, ⋆) of a semihypergroup (H, •) ∈ I . Moreover, we find sufficient conditions for (H, •) and (H+, ⋆) to be cyclic.

Cited by 6 publications

References 14 publications

Semihypergroups obtained by merging of 0-semigroups with groups

Semihypergroups obtained by merging of 0-semigroups with groups

On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations

On Hypergroups with a β-Class of Finite Height

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