Algebraic Structure in a Family of Nim-like Arrays

We study aspects of the algebraic structure shared by a certain family of recur-sively 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 prove that the only loop homomorphisms between loops in the family are either trivial or an identity map