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.