About one special inversion matrix of 3-ary and 4-ary IP-loops

Abstract

It is known that an n-IP-quasigroup can have more than one inversion matrix. We prove that one of these matrices for a 3-ary IP-loop and for a 4-ary IP-loop is the matrix of permutations every of which fixes identity of a loop and has order two. It is a good matrix allowing to investigate n-IP-loops in more detail.