quotient structure

Let $\mathrm{\Sigma }$ be a fixed signature, $𝔄$ a structure for $\mathrm{\Sigma }$, 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 $\{[[a]]\mid a\in 𝔄\}$.

2. For each constant symbol $c$ of $\mathrm{\Sigma }$, $c^{𝔄/\sim }=[[c^{𝔄}]]$.

3. For every natural number $n$ and every $n$-ary function symbol $F$ of $\mathrm{\Sigma }$,

   $$F^{𝔄/\sim }([[a_1]],\mathrm{\dots }[[a_n]])=[[F^{𝔄}(a_1,\mathrm{\dots }{a}_n)]].$$

4. For every natural number $n$ and every $n$-ary relation symbol $R$ of $\mathrm{\Sigma }$,$R^{𝔄/\sim }([[a_1]],\mathrm{\dots },[[a_n]])$ if and only if for some $a_i^{\prime }\sim a_i$we have $R^{𝔄}(a_1^{\prime },\mathrm{\dots },a_n^{\prime }).$