Semi-symmetric isotopic closure of some group varieties and the corresponding identities

Four families of pairwise equivalent identities are given and analyzed. Every identity from each of these families defines one of the following varieties: 1) the semi-symmetric isotopic closure of the variety of all Boolean groups; 2) the semi-symmetric isotopic closure of the variety of all Abelian groups; 3) the semi-symmetric isotopic closure of the variety of all groups; 4) the variety of all semi-symmetric quasigroups. It is proved that these varieties are different and form a chain. Quasigroups belonging to these varieties are described. In particular, quasigroups from 1) and 2) varieties are medial and in addition, they are either groups or non-commutative semi-symmetric quasigroups.