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I am wondering what is the accepted version of the strong order property in continuous logic.

The definition for classical logic is as follows: $T$ has SOP$_n$ (for $n\geq 3$) if there is a formula $\varphi(x,y)$ (where $x$ and $y$ are tuples of the same length) and a sequence $(a_i)_{i<\omega}$ (in some large model of $T$) such that

  1. $\varphi(a_i,a_j)$ holds for all $i<j$;
  2. $\neg\exists x_1\ldots x_n(\varphi(x_1,x_2)\wedge\ldots\wedge\varphi(x_{n-1},x_n)\wedge\varphi(x_n,x_1))$.



The most direct guess for continuous logic would be to just replace "$\varphi(x,y)$ holds" with "$\varphi(x,y)=0$".

$T$ has SOP$_n$ (for $n\geq 3$) if there is a formula $\varphi(x,y)$ (where $x$ and $y$ are tuples of the same length) and a sequence $(a_i)_{i<\omega}$ (in some large model of $T$) such that

  1. $\varphi(a_i,a_j)=0$ for all $i<j$;
  2. $\inf_{x_1,\ldots,x_n}\max\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}>0$.


I am concerned that is translation is too direct and not "approximate" enough. Although it does admit the same characterization as in classical logic of being able to make directed cycles out of the 2-type of an indiscernible sequence:

The following are equivalent.

  1. $T$ has SOP$_n$.
  2. There is an indiscernible sequence $(a_i)_{i<\omega}$ such that if $p(x,y)=\textrm{tp}(a_0,a_1)$ then $$ p(x_1,x_2)\cup\ldots\cup p(x_{n-1},x_n)\cup p(x_n,x_1) $$ is unsatisfiable.

This statement works equally well in the classical or continuous setting (with the definitions above).



If this continuous definition for SOP$_n$ is acceptable, then it seems like the most reasonable way to define the strong order property is just to say:

$T$ has the strong order property if there is a formula $\varphi(x,y)$ (where $x$ and $y$ are tuples of the same length) and a sequence $(a_i)_{i<\omega}$ (in some large model of $T$) such that

  1. $\varphi(a_i,a_j)=0$ for all $i<j$;
  2. for all $n\geq 3$, $\inf_{x_1,\ldots,x_n}\max\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_n,x_1)\}>0$.



So my questions are:

  1. Are these definitions for the strong order property good?
  2. If not, what is the right definition and why don't the ones above work?
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