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Thread: Special Property?

  1. #1
    Senior Member Sampras's Avatar
    May 2009

    Special Property?

    Is there any name given to this: Suppose $\displaystyle \langle S, * \rangle $, $\displaystyle \langle S', *' \rangle $ and $\displaystyle \langle S'', *'' \rangle $ are binary structures. Suppose there is an isomorphism $\displaystyle \phi $ between $\displaystyle S $ and $\displaystyle S' $. Also suppose $\displaystyle S $ and $\displaystyle S' $ are isomorphic. Also $\displaystyle S' $ and $\displaystyle S'' $ are isomorphic, and the same isomorphism $\displaystyle \phi $ are used.
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  2. #2
    MHF Contributor Matt Westwood's Avatar
    Jul 2008
    Reading, UK
    By definition, an isomorphism is a function.

    Two functions are identical only if the domains and ranges are the same.

    So unless $\displaystyle S = S' = S''$ it's not possible for the same isomorphism to be used for $\displaystyle \phi: S \to S'$ as it is for $\displaystyle \phi: S' \to S''$ because by definition the two isomorphism would need to be different.

    As to whether there's a name for this or not, I haven't a clue, but unless there are some unspoken assumptions (or it's a trick question), I'd question what's going on here.
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