Algebraic structure in a family of Nim-like arrays

We study aspects of the algebraic structure shared by a certain family of recursively generated arrays related to the operation of Nim-addition. We first observe that each individual array represents a countably infinite, commutative loop (in the sense of quasigroups). We then prove that each loop in the family is monogenic (generated by a single element in a non-associative fashion), and use this to determine all loop homomorphisms between members of the family.