About the simplifiable cyclic semihypergroups (Q2706767)

A semihypergroup \((H,\;)\) is left (right) cancellative if for all \(a\), \(x\), \(y\) in \(H\): \(ax\cap ay\neq\emptyset\) (\(xa\cap ya\neq \emptyset\)) \(\Rightarrow x=y\). \textit{F. Marty} [8. Skand. Mat.-Kongr. 45-49 (1935; Zbl 0012.05303)] proved that every left (right) cancellative hypergroup is a group. Later on a simpler proof of this result was given by \textit{M. Koskas} [J. Math. Pures Appl., IX. Sér. 49, 155-192 (1970; Zbl 0194.02201)] who asked if every left (right) cancellative semihypergroup is a semigroup. The first example of a cancellative semihypergroup which is not a group was given by \textit{M. Guţan} [Simplifiable cyclic semihypergroups, Proc. Conf. Algebra, Timişoara 1986, 47-50 (1987)].NEWLINENEWLINENEWLINEIn the first part of this paper some results on monogenic cancellative semihypergroups are established. The results concern the complete or reflexive parts of semihypergroups of these kinds.NEWLINENEWLINENEWLINEThe second part of the paper concerns the relation \(\beta\) in semihypergroups. A new proof (using an idea of \textit{I. G. Rosenberg} [Ital. J. Pure Appl. Math. 4, 93-101 (1998; Zbl 0962.20055)]) of a result of \textit{M. Guţan} [Rend. Circ. Mat. Palermo, II. Ser. 45, No. 2, 189-200 (1996; Zbl 0892.20040)] characterizing semihypergroups for which the relation \(\beta\) is transitive is given.