MATH.17:4.2 - Construct the admissible collections

For each relevant pair A, B, specify Adm(A,B), the collection of operations allowed from A to B. Give a condition that can be used to establish membership. Examples include preserving an order, maintaining a relation between components, or mapping a designated subset into itself.

When the operations are functions, MATH.16 supplies the function object and evaluation ev(f,x)=f(x). Restrict that object by the required condition. A rule with several inputs can be represented by a function on their product when a tuple contains all the inputs it needs.

For a partial operation, include its domain of definition. If f is defined on D within A and g on E within B, their composite is defined on {x in D | f(x) in E}. Whether this is an acceptable domain belongs to the current question.

Changing the admissibility condition changes the collection. An update preserving a set of possible states and a map preserving an algebraic operation answer different requirements. State the requirement before using either as an admissible rule.