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Let $S$ be any discrete set. Let $\prec$ be a binary relation defined over $S$. I refer to the corresponding partial order set as $\langle S, \prec \rangle$, which, I presume, being one possible standard nomenclature.

Now, I would like to formally introduce a new binary relation $\vdash$ which is defined only between couples $i,j \in S$ such that $i\prec j$.

Therefore $i \nprec j$ means that $\vdash$ is not defined between $i$ and $j$

I might define the subset $\{(i,j) \in S\times S : i \prec j\}$, but then I don't know how to keep things formal and standard.

My question is: is there a standard syntax of treating this scenario?

Daniele Cuomo
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    What is wrong with simply saying $\vdash \subseteq \prec$, as $\vdash$ is a subset of $\prec$ (other than the obvious weirdness of having three relation symbols in a row, could introduce variables to denote $\vdash$ and $\prec$) – Mor A. Jul 19 '21 at 16:07
  • It could be, but, is it consistent the equivalence class formalism? I don't think so. For instance, I can't write $[i]_{\vdash}$ for any $i \in S$, because $\vdash$ might be undefined for $i$. – Daniele Cuomo Jul 20 '21 at 10:17

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It's simply an equivalence class. https://en.wikipedia.org/wiki/Equivalence_class

egglog
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  • Yes it is, but will be a set $[(i,j)]\subseteq S^2$. In other words, the relation is between two (in general) different sets $X,Y \subseteq S$, therefore I don't know how to refer to the corresponding partial order set – Daniele Cuomo Jul 19 '21 at 16:03