MATH.1:4 - Solution
Local mantra: name the endpoints; form paths; join matching paths; use the composition laws; retain the distinction needed by the next operation.
MATH.1:4.1 - Choose objects and generating arrows
Write the objects at which the elementary steps start and end. For each generator a, give a source s(a) and a target t(a). Several generators may have the same source and target.
Include conditions that determine the availability of the selected continuations. A location may be enough for one problem; another may require a pair such as (location, permission). Use a failed attempted continuation to find the needed distinction. There is no requirement to anticipate every possible future operation.
The generating arrows and endpoints form a directed graph in the mathematical sense. A drawing can display it, but the vertices, arrows and endpoint functions define the structure. A physical network or an organization’s work may be interpreted through this graph; that application supplies its own claims about what the arrows can represent.
MATH.1:4.2 - Form finite paths
A path of length n>0 is a list [a1,…,an] with t(ai)=s(a(i+1)) for every adjacent pair. Its source is s(a1) and its target is t(an).
At each object x, add an empty path id_x. It starts and ends at x and contains no generator. Keeping its object matters: the empty path at one object cannot serve as the identity at another.
Initially, two nonempty paths are equal when their lists contain the same generators in the same order. The empty paths are equal only at the same object. This choice retains the sequence used to construct the path. A later identification can deliberately forget part of it, using MATH.2.
Construct only the paths needed for the immediate question, or describe a family by its formation rule. Cycles make infinitely many paths possible, but each path remains finite.
MATH.1:4.3 - Define composition by concatenation
For paths p:x→y and q:y→z, write p;q for the path obtained by putting the list of q after the list of p. Here the semicolon is read in execution order: first p, then q.
Composition is defined when the full intermediate object agrees. Its source is x and its target is z. If the endpoints disagree, repair the proposed sequence, supply a connecting arrow that is actually permitted, or return the failed connection.
Concatenation gives two laws:
id_x;p=p=p;id_yforp:x→y, because an empty list adds no generator.(p;q);r=p;(q;r)for three consecutively composable paths, because both sides contain the same three lists in the same order.
These arguments establish identity and associativity for this construction. They can be reused while the definitions stay unchanged. They leave the order of the generators intact: p;q and q;p can differ or one can be undefined. A structure with objects, arrows and these laws is a category; the path construction gives the free category on the generating graph.
MATH.1:4.4 - Use the path at the needed level
Return the path that answers the immediate composition question, or the formation and composition rules when the receiving work needs a reusable family.
If each generator has a supplied additive cost, obtain a path’s cost by adding the costs of its generators; the empty path has cost zero. Keep the path as well when the receiver needs to execute it or inspect why it is available. Two paths with the same cost can contain different generators.
If a representation hides an intermediate object, test the attempted composition that made the distinction matter. Refine the objects or retain the paths until the subsequent operation is well defined. MATH.2 constructs an identification that preserves selected operations; it can later reduce the structure deliberately.
An interpretation can associate generators with transformations in another setting. The mathematical path then specifies their composition. Whether those transformations are available and adequate in that setting is the application’s question, using such common methods as FPF C.29 and B.5.MPC.
Stop when the receiving question has a permitted path, a reusable construction, or a specific failed connection. Constructing every possible path or proving an unchanged concatenation law again adds no result to that use.