Hypergroups of type \(U\) on the right of size five. II. (English) Zbl 1187.20070

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

Mat. Vesn. 60, No. 1, 23-45 (2008).

Summary: The hypergroups \(H\) of type \(U\) on the right can be classified in terms of the family \(P_1=\{1\circ x\mid x\in H\}\), where \(1\in H\) is the right scalar identity. If the size of \(H\) is 5, then \(P_1\) can assume only 6 possible values, three of which have been studied in part I [Far East J. Math. Sci. (FJMS) 26, No. 2, 393-418 (2007; Zbl 1142.20053)].
In this paper, we completely describe other two of the remaining possible cases: a) \(P_1=\{\{1\},\{2,3\},\{4\},\{5\}\}\); b) \(P_1=\{\{1\},\{2,3\},\{4,5\}\}\). In these cases, \(P_1\) is a partition of \(H\) and the equivalence relation associated to it is a regular equivalence on \(H\). We find that, apart of isomorphisms, there are exactly 41 hypergroups in case a), and 56 hypergroups in case b).

MSC:

20N20

Hypergroups

Keywords:

hypergroups of type \(U\) on the right; right scalar identities; regular equivalences

Citations:

Zbl 1142.20053