Let $\Sigma$ be a fixed signature, $\A$ a structure for $\Sigma$ and $\sim$ a congruence on $\A$ The quotient structure of $\A$ by $\sim$ denoted $\A/\!\sim$ is defined as follows:

1. The universe of $\A/\!\sim$ is the set $\set{\eqclass{a} \mid a \in \A}$
2. For each constant symbol $c$ of $\Sigma$ .
3. For every natural number $n$ and every $n$ ary function symbol $F$ of $\Sigma$ $$ F^{\A/\!\sim}(\eqclass{a_1}, \ldots \eqclass{a_n}) = \eqclass{F^\A(a_1, \ldots a_n)}. $$
4. For every natural number $n$ and every $n$ ary relation symbol $R$ of $\Sigma$ if and only if for some we have