ways to indicate commutativity and non-commutativity of a diagram when using xymatrix

I used the following code, but \circlearrowleft looks no good in this diagram. I would like to know if there is a better way to indicate the commutativity. Also, is there any way to indicate non-commutativity?

\begin{equation*}
\xymatrix@-1.75pc{
\overline{A} \ar[dd]_{\overline{F}_q}  &  &  \overline{A}_E
\ar[dd]^{\overline{F}_{q,E}} \ar[ll]_{\pi^{\ast}}  \\
& \circlearrowleft & \\
\overline{A}  & &  \overline{A}_E \ar[ll]^{\pi^{\ast}}
}



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Could you explain a bit more what you don't like about \circlearrowleft in the diagram? It may be a bit ugly but it is the usual way to emphasise commutativity. As for non-commutativity then I agree with salvor7 below that the default assumption is that the diagram commutes and I don't know of a standard way of indicating otherwise. One option would be to ask on Maths-SX if there is a standard and then if one emerges to ask how to typeset it here. When I've had to do a non-commuting diagram I've used linguistics as salvor7 says: "We have the following (non-commuting!) diagram". –  Loop Space Feb 21 '13 at 9:39

These map diagrams are usually used to display commutative diagrams, like when a writer writes "the following diagram commutes." Writing "the following diagram commutes" may solve your circle arrow problem by just doing away with the arrow.

These diagrams often called commutative diagrams even when the diagram does not commute. I cannot recall an example where a diagram was shown that did not commute. I think the tendency is to only write down diagrams which do commute, so avoiding non-commutative diagrams might be the right fix. However, you might get away with writing just before the diagram that "the following diagram does not commute," where the emphasis is given to alert the reader that you are departing from the typical.

Also, you might find a fifth map which is an endomorphism (a map from something to itself), \overline{A} in this case, that completes a pentagonal commutative diagram.

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You might want to show non-commutative diagram, for example, to say that one has to be careful not to think that it is commutative. And you might not to write down a phrase for this, for instance, on a presentation where the space is limitd.

One symbol that one could try, for instance, is \ncirclearrowdown from MnSymbol negated arrow. (according to http://ctan.uib.no/info/symbols/comprehensive/symbols-a4.pdf , p.44)

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Welcome to TeX.sx! I marked your inline code with backticks. –  mafp Apr 16 '13 at 15:26