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An automorphism is an isomorphism of an algebraic structure with itself.

Let's review an isomorphism $i$ of two linearly ordered groups $G(+,<)$ and $F(*,<)$.

$i$ is linear order-preserving when $a < b \iff i(a) < i(b)$ for any elements $a$ and $b$ in $G$.

Let's note that every linear order induces the reverse order: $G(+, <, >)$, $F(*, <, >)$.

An isomorphism $i^*$ is linear order-reversing when $a < b \iff i(a) > i(b) \iff i(b) < i(a)$.

Any linear order-preserving isomorphism $i$ induces linear order-reversing isomorphism $i^*$.

In case $G$ and $F$ are different, choosing between $<$ and $>$ in each of the groups is an arbitrary decision, so one can say $i$ and $i^*$ are equivalent.

It looks like this is not the case when we talk about automorphisms.

Let's assume $G$ is commutative and review the inversion automorphism $i(a) = -a$.

$i$ is order-reversing since $a < b \iff -b < -a$.

But in this case we can clearly see the difference between order-preserving and order-reversing automorphisms: $i$ maps $<$ to $>$ on the same set.

I am wondering if it make sense to explicitly say if an automorphism keeps operations and relations the same?

I've never thought of mapping $+$ to $\times$ in an automorphism of an algebraic structure $(+, \times)$.

Are there practical examples of such mapping?

There was a similar discussion with no positive conclusion in here: https://www.researchgate.net/post/Difference_between_an_order_automorphism_and_order_isomoprhism

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  • $\begingroup$ I don't do order theory, but I think that an "order-reversing" isomorphism $(G,<_G)\to (H,<_H)$ is better thought of as an isomorphism to the group $(H,>_H)$, or perhaps as an isomorphism from the domain $(G,>_G)$, much in the same way as we view a contravariant functor $F:C\to D$ as being actually a functor from $C^{\text{op}}$ to $D$. In this light, order reversing isomorphisms between two are not isomorphisms between the the initial objects but isomorphisms between different objects. Thus automorphisms of $(G,<_G)$ should be order-preserving under this viewpoint. $\endgroup$ – jgon Apr 4 at 21:44
  • $\begingroup$ Please don't use $...$ for words in italics, such as $order-preserving$. The correct markup is *order-preserving*. Using $ for math symbols such as $i$ is good, of course. $\endgroup$ – egreg Apr 4 at 22:04
  • $\begingroup$ @egreg: Got it. Thank you. $\endgroup$ – Alex C Apr 4 at 23:06

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