University of Guilan Journal of Algebra and Related Topics 2345-3931 11 2 2023 12 01 On enumeration and classification of $EL^2$-semihypergroups and $EL^2$-$H_v$-semihypergroups with 2 elements 85 97 7141 10.22124/jart.2023.23376.1468 EN S. Mirvakili Department of Mathematics, Payame Noor University, P.O.BOX 19395-3697 Tehran, Iran, S. H. Ghazavi Department of Mathematics, Shahid Ashrafi Esfahani University, Isfahan, Iran M. A. Dehghanizadeh National University of Skills Journal Article 2022 12 04 $EL$-hypergroups were defined by Chvalina 1995. Till<br />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.<br />In this paper, we classify all $EL$-(semi)hypergroups over sets with<br />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. Ends lemma $EL$-hypergroups $H_v$-group quasi order relation partially order relation https://jart.guilan.ac.ir/article_7141_17803814397e52e8fd816245ca91a3fa.pdf