MATH.5:11 - SoTA-Echoing

Question: how can a generator assignment determine an operation-preserving map, including when the source equates different constructions?

Burris and Sankappanavar’s A Course in Universal Algebra, corrected 2012 edition, II §10, definitions 10.1-10.5, lemma 10.6 and theorem 10.8, supplies term formation, recursive evaluation and unique extension. Adopt that construction. The examples apply it to arithmetic expressions and composed affine transformations. The pattern’s quotient step uses the congruence construction supplied by MATH.2 and gives its preservation argument.

The Mathlib free-monoid implementation gives an executable formalization of the word case. FreeMonoid.lift extends generator values by the product of their images; hom_eq states uniqueness from generator agreement. Adapt that presentation into ordinary mathematical instructions, retaining the monoid premises.

The Mathlib path-category construction formalizes the typed branch: Paths.lift extends an object-and-generator assignment; lift_nil, lift_cons and lift_unique establish the identity, recursive composition and uniqueness clauses. The pattern spells out those operations for use after MATH.1. The integer/pair case derives a consequence and a failed additional equation directly from its assigned functions.

The useful comparison is with a supplied map or independent evaluation of a small number of expressions. Extension by generators improves repeated compositional use, while an imposed equation adds a real preservation obligation. Reopen when a partial operation has a domain beyond the path-endpoint construction supplied here, the source admits infinite constructions, an equation fails, or a simpler available map supplies the receiving result.