Parastrophe of Some Inverse Properties in Quasigroups

This work investigates the relationship between the parastrophes of some notion of inverses in quasigroups. Our findings reveal that, of the 5 parastrophes of LIP quasigroup, (23) parastrophe is a LIP quasigroup, (12) and (132) parastrophes are RIP quasigroups, while (13) and (132) parastrophes are an anti-commutative quasigroup. Similarly, the (12) and (132) parastrophes of a RIP quasigroup are LIP quasigroups; the (13) parastrophe of a RIP quasigroup is a RIP quasigroup, while the (23) and (123) parastrophes are anti-commutative quasigroups. As for the CIP quasigroup, only the (12) parastrophe is a CIP quasigroup; other parastrophes are symmetric quasigroups of order two. Finally, the (12) parastrophe of the WIP quasigroup is an IP quasigroup, the (13), (23), and (132) parastrophes of the WIP quasigroup are CIP quasigroups, while the (123) parastrophe of the WIP quasigroup is a WIP quasigroup.