Let $S$ be a set. Let $\sim$ be a binary relation on $S$. Suppose $\sim$ follows these three rules.
- $x\sim x$ for all $x\in S$ (reflexivity).
- If $x\sim y$, then $y\sim x$ for all $x, y \in S$ (symmetry).
- If $x_1\sim x_2 $, $x_2\sim x_3 $, $x_3\sim \cdots$, then there exists an $n>2$ in the indices of this chain of similarities such that $x_1\nsim x_n$.
I'm trying to capture kind of "partial" or "fuzzy" transitivity. I'm not sure if these are quite the right rules and I'd be happy for anyone to refine them. It reminds me of that children's game, "Whisper Down the Lane", where children in a row whisper a message into each others' ears. Eventually, the final message does not resemble the initial one at all, nor probably a few others.
Update: See comments for modifications on this definition.
These rules seem to be inconsistent. For example, $x\sim x\sim x\sim x\cdots$, so the third rule cannot hold in this case.
A relation that seems to meet what you want, on $\mathbb{R}$, is $D_h(x,y)$ where $xD_hy$ if and only if $|x-y|\le h$. It doesn't meet the third condition, but it models the children's game.