I'm trying to understand some notation which is unfamiliar to me and I am struggling to see the logic behind it.
I gather that the use of $[x]_\rho$ represents the set of $\rho$ equivalence classes on S for example (where $\rho$ is a relation). But what I have come across involves $[x]_J$ where J is an ideal of S.
I don't understand how this works, does the subscript not need to be a relation?
The link below shows what I am trying to follow, namely the solution to excercise 1.20 on page 275 (or 284 of the document).
Each ideal $J$ of a semigroup defines a relation $\rho_J$ defined by $$ {\rho_J} = (J \times J) \cup (J^c \times J^c). $$ In other words $s \mathrel{\rho_J} t$ if and only if $s$ and $t$ are simultaneously in $J$ or simultaneously in $J^c$. The notation $[x]_J$ is just a shortcut for $[x]_{\rho_J}$.