MATH.5:1 - Problem frame

Use this pattern when you know what the basic elements of a mathematical construction should become and need a compatible map on everything built from them. A value assigned to a variable, a transformation assigned to a command, or a cost assigned to a generating step must extend to composite expressions in a way that preserves the chosen operations.

The needed map is a homomorphism: it carries each source operation to the corresponding target operation. To construct it, evaluate composite expressions from the assigned generator values. If the source identifies different expressions, establish that those expressions receive the same value.

Start with one generator and one composite expression. Return their images and the rule that extends the assignment, or a source equality that the proposed images cannot preserve. A first calculation can expose an incompatible assignment before a large translation is attempted.

The reader needs functions, finite expressions and the operations used in the chosen construction. The method covers total operations with finitely many inputs, finite words with associative composition, and the finite typed paths built by MATH.1. A typed path starts and ends at named objects; its interpretation must supply their images as well as the images of its arrows.

If the required map or one needed value is already available, use it. An arbitrary map on a small finite set can be simpler to specify directly when preserving operations is not part of the question.