I need an interpretation for the expression $\phi / \theta$ where $\phi$ and $\theta$ are congruence relations which satisfy $\theta \subsetneq \phi$.
I know for a set $M$ and a Relation $R$ with $M/R$ is the quotientspace or in other words the set of all equivalence classes. But I'm not sure how the Relation "acts" on other 2-tupels.
Some help would be nice!
On
Just found a lemma in an algebra book which solves my problem:
Let $\phi$, $\theta$ be congruence relations on an algebra $\mathfrak{A}$ with $\theta \subset \phi$. Then the relation $\phi / \theta := \{([a]_{ \theta },[b]_{\theta})\vert(a,b) \in \phi\}$ is a congruence relation on $\mathfrak{A}/\theta$.
I hope this simple example will help you to understand the general case. Consider on $\mathbb{Z}$ the equivalence relations $R_2$ and $R_6$ defined by \begin{align} x \mathrel{R_2} y &\iff x \equiv y \bmod 2 \\ x \mathrel{R_6} y &\iff x \equiv y \bmod 6 \end{align} Clearly $R_6 \subset R_2$. Now, $\mathbb{Z}/R_6 = \mathbb{Z}/6\mathbb{Z} = \{0, 1, 2, 3, 4, 5\}$. The relation $R_2/R_6$ is the relation induced on $\mathbb{Z}/R_6$ by the relation $R_2$ on $\mathbb{Z}$: its two classes, namely $\{0, 2, 4\}$ and $\{1, 3, 5\}$ are obtained by considering the classes modulo $6$ of the sets of even (respectively odd) integers.