ABSTRACT
This chapter discusses various algebraic concepts associated with quasigroups: substructures, homomorphisms, and products. Principal isotopy clarifies the relationship between the various quasigroups obtained with a given Latin square as the body of their multiplication table. There are various approaches to the construction of Latin squares. By definition, quasigroups satisfy the property ensuring that the body of a finite group multiplication table is a Latin square. Thus the body of the multiplication table of a finite quasigroup will also be a Latin square. Quasigroups are defined so that division is always possible. In fact, there are two forms of division in a quasigroup: from the left, and from the right. Just like rings, groups, and semigroups, quasigroups are abstract algebras in their own right. As such, they come equipped with their corresponding concepts of substructure, homomorphism, and product. Subquasigroups of a quasigroup are characterized by closure under the three operations of multiplication, left division, and right division.
