PlanetMath: quotient structure

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quotient structure

(Definition)

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:

The universe of $\mathfrak{A}/\!\sim$ is the set $\{[\![a]\!] \mid a \in \mathfrak{A}\}$ .
For each constant symbol of $\Sigma$ , $c^{\mathfrak{A}/\!\sim} = [\![c^\mathfrak{A}]\!]$ .
For every natural number and every -ary function symbol of $\Sigma$ ,
$\displaystyle F^{\mathfrak{A}/\!\sim}([\![a_1]\!], \ldots [\![a_n]\!]) = [\![F^\mathfrak{A}(a_1, \ldots a_n)]\!].$
For every natural number and every -ary relation symbol of $\Sigma$ , $R^{\mathfrak{A}/\!\sim}([\![a_1]\!], \ldots, [\![a_n]\!])$ if and only if for some $a_i' \sim a_i$ we have $R^\mathfrak{A}(a_1', \ldots, a_n').$

"quotient structure" is owned by almann.

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Cross-references: relation symbol, function symbol, natural number, constant symbol, universe, congruence, structure, signature, fixed
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This is version 7 of quotient structure, born on 2003-07-20, modified 2003-09-16.
Object id is 4487, canonical name is QuotientStructure.
Accessed 1946 times total.

Classification:

AMS MSC:	03C05 (Mathematical logic and foundations :: Model theory :: Equational classes, universal algebra)
	03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)

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