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$EL$-hypergroups  were defined by Chvalina 1995. Till now, no exact statistics of $EL$-hypergroups have been done. Moreover, there is no classification of $EL$-hypergroups and $EL^2$-hypergroups even over small sets. In this paper we classify all $EL$-(semi)hypergroups over sets with two elements obtained from quasi ordered semigroups. Also, we characterize all quasi ordered $H_v$-group and   then we enumerate the number of $EL^2$-$H_v$-hypergroups and $EL^2$-hypergroups of order $2$.