On Commutativity and Mediality of Polyagroup Cross Isomorphs

The notion of isotopy (cross isomorphism) of $n$-ary operations can be got from the well-known notion of isotopy (isomorphism) by replacing one of its components with a $k$-ary $m$-invertible operation [1, 2]. The idea of consideration of cross isotopy belongs to V. D. Belousov [3], who defined it for binary quasigroups.
In the paper necessary and sufficient conditions for commutativity and mediality of a polyagroup cross isomorph (when $n>2k$) are determined. A neutrality criterion of an arbitrary element is stated.