Structure-preserving maps are mathematical functions that maintain certain properties between two structures, such as groups, rings, or topological spaces. These maps ensure that the relationships and operations defined in one structure are reflected in the other, allowing for meaningful comparisons and transformations. For example, a homomorphism is a type of structure-preserving map between two algebraic structures like groups or rings. It preserves operations, meaning if you apply the operation in the first structure, the result will correspond to the operation in the second structure, thus maintaining the underlying structure's integrity.