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.
This contribution deals with
n
-ary relations and hyperstructure theory. There exists a way of creating semihypergroups and hypergroups from (partially) quasi-ordered (semi)groups known as Ends lemma ...construction. In this paper, we use this method to introduce a new class of (semi)hypergroup from a given (semi)group endowed with a preordering
n
-ary relation as a generalization of EL-hyperstructures. Then, we study some basic properties and important elements belonging to this class and the essential differences between this new class and the earlier one (i.e. EL-hyperstructures) are also investigated.
In the paper we study ordered sets in which multi-valued aspects are important. To be more precise, we focus on
EL
-hyperstructures, i.e. semihypergroups constructed from partially ordered ...semigroups. We show that when certain slight modifications of the concept (motivated by real-life situations) are introduced, the applicability of this model increases as we are able to reduce the number of its assumptions, bypass some existing problems and at the same time apply the idea in a greater number of contexts.
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