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I am copying my question from here: https://math.stackexchange.com/q/3233462/427611.

Is it correct that $\mathbb Z/3\mathbb Z$ and $\mathbb Z/4\mathbb Z$ are the only rings with three or more elements with a non-linear cyclic order compatible with both the addition and the multiplication?

A cyclic order is compatible with addition if $[a, b, c] \implies [a + x, b + x, c + x]$ and $[x + a, x + b, x + c]$ for any elements $a$, $b$, $c$, $x$ of the ring.

A cyclic order is compatible with multiplication if $[a, b, c] \implies [ax, bx, cx]$ and $[xa, xb, xc]$ for any elements $a$, $b$, $c$, and any positive element $x$ of the ring.

A element $x$ of a ring with a cyclic order is positive if $[0, x, -x]$.

A cyclic order on a ring is non-linear if any cut of it is not compatible with the addition or multiplication.

A cut of a cycle order on a ring is a linear order $<$ such that $a < b < c \implies [a, b, c]$ for any elements $a$, $b$, $c$ of the ring.

A cut of a cyclic order is compatible with addition if $a < b \implies a + x < b + x$ and $x + a < x + b$ for any elements $a$, $b$, $x$ of the ring.

A cut of a cyclic order is compatible with multiplication if $a < b \implies ax < bx$ and $xa < xb$ for any elements $a$, $b$, and any positive element $x$ of the ring.

An element $x$ of a cut of a cyclic order is positive if $0 < x$.

Below is a sketch of the proof provided on the referenced page:

  1. A non-linearly cyclically ordered group has four quadrants;
  2. If a non-linearly cyclically ordered group has more than one positive element, then all the four quadrants are not empty;
  3. The multiplication of elements from different quadrants is incompatible with the cyclic order.

One can find the details of my research on the item in here:

  1. Cycle notation for cyclic orders: https://math.stackexchange.com/q/3236651/427611

  2. Positive and negative elements of a cyclically ordered group: https://math.stackexchange.com/q/2213048/427611

  3. Apex of a cyclically ordered group: https://math.stackexchange.com/q/2204247/427611

  4. Quadrants of a cyclically ordered group: https://math.stackexchange.com/q/3230720/427611

  5. Natural cut of a cyclically ordered group: https://math.stackexchange.com/q/3207182/427611

  6. Compatibility with multiplication of a cyclic order on a ring: https://math.stackexchange.com/q/3233462/427611

  7. A property of an Archimedean cyclically ordered group: https://math.stackexchange.com/q/2205470/427611

  8. The rule of three steps for a cyclically ordered group: https://math.stackexchange.com/q/2208205/427611

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  • $\begingroup$ I think it would be useful to provide less but better focussed and brief information. I added (from your MathSE post) the main definition. $\endgroup$
    – YCor
    Commented May 2, 2020 at 23:11
  • $\begingroup$ @YCor I added a sketch of the proof. Would it be sufficient? $\endgroup$
    – Alex C
    Commented May 2, 2020 at 23:23
  • $\begingroup$ At least it clarifies the intent of your question (which is a proof verification). $\endgroup$
    – YCor
    Commented May 2, 2020 at 23:25
  • $\begingroup$ @YCor Or a shorter proof if possible. $\endgroup$
    – Alex C
    Commented May 2, 2020 at 23:28
  • $\begingroup$ Actually I don't understand the definition: what is meant by "positive" in a cyclic order? $\endgroup$
    – YCor
    Commented May 2, 2020 at 23:31

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