quotient structure

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

1.

The universe of $\mathfrak{A}/\!\sim$ is the set $\{[\![a]\!]\mid a\in\mathfrak{A}\}$ .
2.

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

For every natural number $n$ and every $n$ -ary function symbol $F$ of $\Sigma$ ,

$F^{\mathfrak{A}/\!\sim}([\![a_{1}]\!],\ldots[\![a_{n}]\!])=[\![F^{\mathfrak{A}% }(a_{1},\ldots a_{n})]\!].$

4.

For every natural number $n$ and every $n$ -ary relation symbol $R$ of $\Sigma$ , $R^{\mathfrak{A}/\!\sim}([\![a_{1}]\!],\ldots,[\![a_{n}]\!])$ if and only if for some $a_{i}^{\prime}\sim a_{i}$ we have $R^{\mathfrak{A}}(a_{1}^{\prime},\ldots,a_{n}^{\prime}).$

Title	quotient structure
Canonical name	QuotientStructure
Date of creation	2013-03-22 13:46:41
Last modified on	2013-03-22 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