# Special Property?

• Aug 30th 2009, 11:09 AM
Sampras
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.
• Aug 30th 2009, 12:03 PM
Matt Westwood
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.