quotient structure
Let $\mathrm{\Sigma}$ be a fixed signature^{}, $\U0001d504$ a structure^{} for $\mathrm{\Sigma}$, and $\sim $ a congruence^{} on $\U0001d504$. The quotient structure of $\U0001d504$ by $\sim $, denoted $\U0001d504/\sim $, is defined as follows:

1.
The universe^{} of $\U0001d504/\sim $ is the set $\{[[a]]\mid a\in \U0001d504\}$.

2.
For each constant symbol $c$ of $\mathrm{\Sigma}$, ${c}^{\U0001d504/\sim}=[[{c}^{\U0001d504}]]$.

3.
For every natural number^{} $n$ and every $n$ary function symbol $F$ of $\mathrm{\Sigma}$,
$${F}^{\U0001d504/\sim}([[{a}_{1}]],\mathrm{\dots}[[{a}_{n}]])=[[{F}^{\U0001d504}({a}_{1},\mathrm{\dots}{a}_{n})]].$$ 
4.
For every natural number $n$ and every $n$ary relation symbol $R$ of $\mathrm{\Sigma}$, ${R}^{\U0001d504/\sim}([[{a}_{1}]],\mathrm{\dots},[[{a}_{n}]])$ if and only if for some ${a}_{i}^{\prime}\sim {a}_{i}$ we have ${R}^{\U0001d504}({a}_{1}^{\prime},\mathrm{\dots},{a}_{n}^{\prime}).$
Title  quotient structure 

Canonical name  QuotientStructure 
Date of creation  20130322 13:46:41 
Last modified on  20130322 13:46:41 
Owner  almann (2526) 
Last modified by  almann (2526) 
Numerical id  10 
Author  almann (2526) 
Entry type  Definition 
Classification  msc 03C05 
Classification  msc 03C07 