Parastrophe of Some Inverse Properties in Quasigroups

This work investigates the relationship that exists between the parastrophes of some notion of inverses in quasigroups. Our findings revealed 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 an RIP quasigroup, while the (23)- and (123)-parastrophes are anti-commutative quasigroups. As for the CIP quasigroup, only (12)- parastrophe is a CIP quasigroup; other parastrophes are symmetric quasigroups of order 2. Finally, (12)-parastrophe of WIP quasigroup is an IP quasigroup, (13)-,(23)- and (132)-parastrophes of WIP quasigroup are CIP quasigroups, while (123)-parastrophe of WIP quasigroup is a WIP quasigroup.