The Clowder Project

In detail, by Chapter 7: Equivalence Relations and Apartness Relations, Construction 7.4.2.1.2, the relation $\mathord {\sim }$ of Definition 2.2.5.1.1 is given by declaring $a\sim b$ iff one of the following conditions is satisfied:

• We have $a=b$;
• There exist $x_{1},\ldots ,x_{n}\in B$ such that $a\sim 'x_{1}\sim '\cdots \sim 'x_{n}\sim 'b$, where we declare $x\sim 'y$ if one of the following conditions is satisfied:
  1. There exists $z\in A$ such that $x=f\webleft (z\webright )$ and $y=g\webleft (z\webright )$.
  2. There exists $z\in A$ such that $x=g\webleft (z\webright )$ and $y=f\webleft (z\webright )$.

  That is: we require the following condition to be satisfied:

  • There exist $x_{1},\ldots ,x_{n}\in B$ satisfying the following conditions:
    1. There exists $z_{0}\in A$ satisfying one of the following conditions:
       1. We have $a=f\webleft (z_{0}\webright )$ and $x_{1}=g\webleft (z_{0}\webright )$.
       2. We have $a=g\webleft (z_{0}\webright )$ and $x_{1}=f\webleft (z_{0}\webright )$.
    2. For each $1\leq i\leq n-1$, there exists $z_{i}\in A$ satisfying one of the following conditions:
       1. We have $x_{i}=f\webleft (z_{i}\webright )$ and $x_{i+1}=g\webleft (z_{i}\webright )$.
       2. We have $x_{i}=g\webleft (z_{i}\webright )$ and $x_{i+1}=f\webleft (z_{i}\webright )$.
    3. There exists $z_{n}\in A$ satisfying one of the following conditions:
       1. We have $x_{n}=f\webleft (z_{n}\webright )$ and $b=g\webleft (z_{n}\webright )$.
       2. We have $x_{n}=g\webleft (z_{n}\webright )$ and $b=f\webleft (z_{n}\webright )$.