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. 1.

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

2. 2.

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

3. 3.

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. 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 QuotientStructure 2013-03-22 13:46:41 2013-03-22 13:46:41 almann (2526) almann (2526) 10 almann (2526) Definition msc 03C05 msc 03C07