 
Summary: Algebraic Structure in a Family of Nimlike Arrays
Lowell Abrams
Department of Mathematics
The George Washington University
Washington, DC 20052 U.S.A.
labrams@gwu.edu
Dena S. CowenMorton
Department of Mathematics
Xavier University
Cincinnati, OH 452074441 U.S.A.
morton@xavier.edu
September 10, 2009
Abstract
We study aspects of the algebraic structure shared by a certain family of recur
sively generated arrays related to the operation of Nimaddition. 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 nonassociative fashion), and use this to determine all loop
homomorphisms between members of the family.
Keywords: quasigroups and loops, monogenic, Nim, SpragueGrundy, sequential com
