The Clowder Project

In detail, by , , the relation $\sim$ 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 $x_1,\dots ,x_n\in B$ such that $a{\sim }^{\prime }{x}_1{\sim }^{\prime }\cdots {\sim }^{\prime }{x}_n{\sim }^{\prime }b$, where we declare $x{\sim }^{\prime }y$ if one of the following conditions is satisfied:
  1. (a) There exists $z\in A$ such that $x=f\left(z\right)$ and $y=g\left(z\right)$.
  2. (b) There exists $z\in A$ such that $x=g\left(z\right)$ and $y=f\left(z\right)$.

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

  • There exist $x_1,\dots ,x_n\in B$ satisfying the following conditions:
    1. (i) There exists $z_0\in A$ satisfying one of the following conditions:
       1. (I) We have $a=f\left(z_0\right)$ and $x_1=g\left(z_0\right)$.
       2. (II) We have $a=g\left(z_0\right)$ and $x_1=f\left(z_0\right)$.
    2. (ii) For each $1\le i\le n-1$, there exists $z_i\in A$ satisfying one of the following conditions:
       1. (I) We have $x_i=f\left(z_i\right)$ and $x_{i+1}=g\left(z_i\right)$.
       2. (II) We have $x_i=g\left(z_i\right)$ and $x_{i+1}=f\left(z_i\right)$.
    3. (iii) There exists $z_n\in A$ satisfying one of the following conditions:
       1. (I) We have $x_n=f\left(z_n\right)$ and $b=g\left(z_n\right)$.
       2. (II) We have $x_n=g\left(z_n\right)$ and $b=f\left(z_n\right)$.