New aspects in polygroup theory

Sonea, Andromeda Cristina

doi:10.2478/auom-2020-0044

Cited by 5 publications

(5 citation statements)

References 5 publications

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“…Indeed, the completeness degree of weakly complete hypergroups admits simple closed formulas. Furthermore, it can be related to the commutativity degree, which has been recently brought into hypercompositional algebra from group theory [13,18].…”

Section: Discussionmentioning

confidence: 99%

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Commutativity and Completeness Degrees of Weakly Complete Hypergroups

et al. 2022

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We introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β-classes.

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Section: Discussionmentioning

confidence: 99%

“…Recently, the concept of commutativity degree has been introduced also in hypergroup theory [13,18]. In particular, in [13] the authors defined the commutativity degree of a finite hypergroup (H, •) as…”

Section: Commutativity Degree Of Weakly Complete Hypergroupsmentioning

confidence: 99%

Commutativity and Completeness Degrees of Weakly Complete Hypergroups

et al. 2022

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“…This is the probability that two randomly chosen elements commute in a non-commutative hypergroup. It has been already studied for HX-groups [25] and polygroups [26]. First, we have presented a general method to compute this degree, obtaining a formula that depends on the commutativity degree of the group that appears in the representation theorem of the considered complete hypergroup.…”

Section: Discussionmentioning

confidence: 99%

The Class Equation and the Commutativity Degree for Complete Hypergroups

Sonea

Cristea

2020

Mathematics

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The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

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“…In this subsection, we will briefly recall the terminology and notations related to polygroups. New properties have been recently investigated by Sonea [39] and Al Tahan et al [40] in connection with the commutativity degree of finite polygroups. This property for complete hypergroups has been investigated in [18].…”

Section: Polygroupsmentioning

confidence: 99%

From HX-Groups to HX-Polygroups

Mousavi,

Jafarpour,

Cristea

2023

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HX-groups are a natural generalization of groups that are similar in construction to hypergroups. However, they do not have to be considered as hypercompositional structures like hypergroups; instead, they are classical groups. After clarifying this difference between the two algebraic structures, we review the main properties of HX-groups, focusing on the regularity property. An HX-group G on a group G with the identity e is called regular whenever the identity E of G contains e. Any regular HX-group may be characterized as a group of cosets, and equivalent conditions for describing this property are established. New properties of HX-groups are discussed and illustrated by examples. These properties are uniformity and essentiality. In the second part of the paper, we introduce a new algebraic structure, that of HX-polygroups on a polygroup. Similarly to HX-groups, we propose some characterizations of HX-polygroups as polygroups of cosets or double cosets. We conclude the paper by proposing several lines of research related to HX-groups.

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New aspects in polygroup theory

Cited by 5 publications

References 5 publications

Commutativity and Completeness Degrees of Weakly Complete Hypergroups

Commutativity and Completeness Degrees of Weakly Complete Hypergroups

The Class Equation and the Commutativity Degree for Complete Hypergroups

From HX-Groups to HX-Polygroups

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