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Commutativity preserves IS

June 24, 2012

Dear All,

We all think of function as a process that transforms one object into another or as an equation or as an operation assigning one element of codomain to each element of domain.  In all these descriptions the emphasis is on ‘change’ that the function brings about.  In addition to changing inputs to outputs, if one is into that terminology, there’s something that the function preserves.  In taking elements to elements the function preserves, if I may say so, element-hood.  It’s not that we don’t know that a function assigns elements to elements, but that all too visible facet of function is not highlighted in our conception of function (in our mind or at least my mind).  We can spin a similar yarn w.r.t composition.  The composite of two functions, when defined, is a function.  In other words, composition preserves function-hood.  We can compose not only functions, but commutative diagrams (triangles) depicting composition.  Composite of 2 commutative triangles, when defined, is a commutative diagram (square); so we may say composition preserves commutativity.  Of course, this is all common knowledge; common knowledge, when it’s too common runs the risk of entering practice without conscious awareness that’s needed to communicate.

We use the above exercise of asserting the all too obvious as warm-up to work-out the relation between commutativity and structure (Conceptual Mathematics, page 136).  For the sake of concreteness, we consider two sets A and B, with endomaps x: A –> A and y: B –> B, representing structure on A and B, respectively.  Now given a map f: A –> B, we want to find the relation between f and what it does to the structure (A, x) as it maps into (B, y).

We’ll look at various functions f (also x, and y) beginning with A and B both as singletons, and going all the way until we find the relation between commutativity (whether or not fx = yf) and structure (A, x) mapped into (B, y).  We’d like to know what f does and does not do to the structure (A, x) as it maps into (B, y) when it satisfies fx = yf i.e. when the diagram commutes and when it does not satisfy fx = yf i.e. when the diagram does not commute. Having mastered the art of attending to the mundane (see above), we should be able to get good grasp of the relation between commutativity and structure by the time we get to the cases of both A and B having 2 elements.  We could make the exercise less abstract by considering the structure on sets A and B as ‘<=’ or even more concrete by way of going numerical, say, A = {2, -2}, B = {4, -4} and considering two functions; f1: a |–>2a and f2: a |–> -2a.

In any case I make it (attached Commutative Diagram) as much fun for you as I possibly could.

Thank you,


Commutative Diagram

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