Tility

Tility refers to a property of mathematical objects, particularly in the context of abstract algebra and category theory, indicating that certain operations or mappings preserve specific structural characteristics. For instance, in group theory, a homomorphism between two groups is said to be tility if it preserves the group operation. This means that for any elements and in the first group, the image of their product under the homomorphism is equal to the product of their images. This concept is crucial for understanding how structures can be related and how information is transferred between them.

In a broader sense, tility can be applied to various mathematical structures beyond groups, such as rings, modules, or vector spaces. A function or operator exhibits tility if it respects the fundamental operations and relations defined within these structures. For example, a linear transformation between vector spaces is a form of tility, as it preserves vector addition and scalar multiplication. The idea is to identify mappings that act as "good citizens" within the algebraic landscape, upholding the inherent properties of the objects they connect. This preservation of structure allows mathematicians to build analogies, prove theorems about related objects, and simplify complex problems by transferring them to more manageable settings.