Hypergroups of type U on the right of size five. Part two

Abstract

The hypergroups $H$ of type $U$ on the right can be classified in termsof the family $P_{1}=\{1\circ x\mid xın H\}$, where $1ın H$ isthe right scalar identity. If the size of $H$ is $5$, then $P_{1}$ canassume only $6$ possible values, three of which have been studied inthe first part of the paper. In this paper, we completelydescribe 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 equivalencerelation associated to it is a regular equivalence on $H$.We find that, apart of isomorphisms, there are exactly $41$hypergroups in case~a), and $56$ hypergroup in case~b).