MATH.5:4 - Solution

Local mantra: name the generators and operations; assign their images; extend by construction; preserve the equations; use the resulting map.

MATH.5:4.1 - State the source construction and target operations

List the generators and the rules for building finite expressions from them. An expression can be a generator, a constant, or an operation applied to constituent expressions. State each operation’s number of inputs. With generators x,y, constants 0,1 and binary operations + and *, the expression (x*y)+1 is obtained by two construction steps.

Distinguish the freely formed expressions from any equations imposed on them. For example, treating x*y and y*x as the same requires a commutativity equation. The written resemblance of the expressions does not supply that equation.

Name the target set and a corresponding operation for each source operation, including the value of each constant. Every required target operation must be defined on the inputs it receives. This pattern’s total-operation construction does not supply a partial operation’s domain.

For a word construction, state the target composition star and its identity e. The operation must be associative for unparenthesized words to denote a composition independently of grouping. MATH.1 constructs the source words or paths; here the task is to build their map into the target structure.

For typed paths, assign each source object X a target object F(X). The target must supply arrows, associative composition on matching endpoints, and an identity at each object. Assign each generator a:X -> Y an arrow a_target:F(X) -> F(Y). A target can consist of sets and functions; different source objects may then receive sets of different kinds of values. Check these domains before composing.

MATH.5:4.2 - Extend the assignment recursively

Assign a target value a(x) to every generator x. Define the evaluation E on finite expressions:

• a generator x receives a(x);
• a constant receives its named target value;
• op(t1,...,tn) receives op_target(E(t1),...,E(tn)).

Each recursive call uses a constituent expression. MATH.4 supplies the finite-construction argument and the treatment of a result that needs additional information.

For a word [x1,...,xk], the same move evaluates the assigned generator values in order. The empty word receives e; extending a word by x changes its value from v to v star a(x). Associativity and the identity laws make this evaluation preserve concatenation:

E(p;q)=E(p) star E(q).

For a typed path, set E(id_X)=id_F(X). If p:X -> Y has been evaluated and a:Y -> Z is the next generator, set E(p;a)=E(p) star a_target, where star is again read in execution order. The intermediate object F(Y) makes this composition defined. Recursing by path length evaluates every finite path while keeping its endpoints. In a target of sets and functions, this means applying E(p) first and a_target second.

The image of a composite is now computed from its parts. It is no longer an independently chosen entry in a correspondence table.

MATH.5:4.3 - Establish preservation and uniqueness

For freely formed expressions, preservation follows from the defining clause: evaluating an operation on expressions gives the target operation on their evaluations. The generator and constant clauses cover the starting cases.

Suppose another operation-preserving map H has the same assigned generator values. It agrees with E on generators and constants. If it agrees on the constituents of an expression, preservation forces agreement on the whole expression. Induction therefore gives H(t)=E(t) for every expression.

The conclusion is uniqueness among maps preserving the named operations and agreeing with the given assignment. Changing the assignment or required operations changes that question.

For words, the corresponding argument starts with the empty word and extends by one generator. Every concatenation-preserving map with the same identity and generator images must return the same ordered product.

For paths, the base case uses the identity at each F(X). Induction on the second path’s length, using target associativity, gives E(p;q)=E(p) star E(q) for every permitted join. A map preserving identities and composition with the same object and generator assignments must follow those recursive clauses, so it is unique. Such an object-and-arrow map between categories is called a functor. The one-object word construction is its monoid case.

MATH.5:4.4 - Make the map respect identified expressions

If the source equates expressions, test the equations under E. To define E_bar([t])=E(t) on an equivalence class, require:

t~u implies E(t)=E(u).

MATH.2 supplies the quotient and representative-independence construction. Here it is applied to the evaluation just obtained.

For typed paths with the objects retained, impose equations between paths having the same source and target. Compare their target arrows, including the appropriate identity for an empty path. Closure under permitted composition makes the evaluation descend just as above. Identifying different source objects changes this setup and requires a construction that also accounts for their identities and permitted joins.

When the source equivalence is generated by stated equations and their use inside larger expressions, show that each generating equation has equal evaluated sides. Equality is preserved when equal values enter the same target operation. It is also preserved along reversal and a finite sequence of equation replacements. These facts extend the result to the generated equivalence.

An equation schema such as s*t=t*s ranges over its permitted substitutions. Checking one numerical substitution leaves the other instances unresolved. Establish the target law for the required range or return a failing instance. If the source also has additional identifications, include them in the comparison.

A failed equation gives a concrete choice: change the generator assignment, change the target operations, or use a source construction that retains the distinction. Each option changes the mathematical account. Select the one that still answers the receiving question.

MATH.5:4.5 - Use the map and inspect what it forgets

Evaluate the needed expression or pass the reusable map to its consumer. The preservation argument permits calculation by parts and supports substitution of an equivalent source expression.

Equal target values can still come from different source objects. If the receiver needs to recover a source object, obtain an inverse, a retained representative, or another construction that supplies it. A homomorphism by itself does not provide such recovery.

After a changed generator value, reuse the recursive definition and preservation proof for freely formed expressions. Recheck any imposed equation whose evaluated sides can change. After a changed target operation, revisit the clauses and laws that used it.

Stop with the needed value, the reusable homomorphism, or an equation that prevents the proposed extension. A mathematical map can then contribute to FPF C.29’s interpretation-and-return method when the receiving question concerns another subject.