Abstract
In the algebra of single-valued structures, cyclicity is one of the fundamental properties of groups. Therefore, it is natural to study it also in the algebra of multivalued structures (algebraic hyperstructure theory). However, when one considers the nature of generalizing this property, at least two (or rather three) approaches seem natural. Historically, all of these had been introduced and studied by 1990. However, since most of the results had originally been published in journals without proper international impact and later—without the possibility to include proper background and context-synthetized in books, the current way of treating the concept of cyclicity in the algebraic hyperstructure theory is often rather confusing. Therefore, we start our paper with a rather long introduction giving an overview and motivation of existing approaches to the cyclicity in algebraic hyperstructures. In the second part of our paper, we relate these to E L -hyperstructures, a broad class of algebraic hyperstructures constructed from (pre)ordered (semi)groups, which were defined and started to be studied much later than sources discussed in the introduction were published.
Highlights
In the algebra of single-valued structures, cyclicity is one of the fundamental properties of groups
The additive group of integers serves as an example of an infinite cyclic group while its decomposition modulo n serves as an example of a finite cyclic group
It is evident that the set of elements generated by a single element of a hypergroup is closed, . . . , but I am unable to say whether or not it forms a hypergroup”, and he sought some analogy of the representation of a cyclic group as the group of rotations of a regular polygon
Summary
Prior to including Wall’s definition, we recall the well-known definition of the (algebraic) hypergroup. We mean a pair ( H, ∗), where “∗” is a mapping H × H → P ∗ ( H ), where P ∗ ( H ) is the set of all non-empty subsets of H. We mean a hypergroupoid which is associative, i.e., for all a, b, c ∈ H, there is a ∗ (b ∗ c) = ( a ∗ b) ∗ c, and reproductive, i.e., for all a ∈ H, there. An associative hypergroupoid is called semihypergroup; a reproductive hypergroupoid is called quasihypergroup. If a hypergroup H is generated by a single element a of H, H will be called a cyclic hypergroup. It is evident that the set of elements generated by a single element of a hypergroup is closed, . It is evident that the set of elements generated by a single element of a hypergroup is closed, . . . , but I am unable to say whether or not it forms a hypergroup”, and he sought some analogy of the representation of a cyclic group as the group of rotations of a regular polygon
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