Wednesday, 16 January 2013

order theory - Prove that, for all ordered sets P, Q and R,



PQRP×QR




where QR is the set of all order-preserving maps from Q to P; is order-isomorphic symbol; × is cartesian product symbol.



Besides I am not good at building bijection to prove isomorphism, I hope you can teach me some technique on this issue.


Answer



Suppose that φPQR; then for each pP, φ(p) is an order-preserving map from Q to R. Define



ˆφ:P×QR:p,q(φ(p))(q).



Show that the map φˆφ is the desired isomorphism.


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