quotient structure

Let Σ be a fixed signaturePlanetmathPlanetmathPlanetmath, 𝔄 a structureMathworldPlanetmath for Σ, and a congruencePlanetmathPlanetmathPlanetmath on 𝔄. The quotient structure of 𝔄 by , denoted 𝔄/, is defined as follows:

  1. 1.

    The universePlanetmathPlanetmath of 𝔄/ is the set {[[a]]a𝔄}.

  2. 2.

    For each constant symbol c of Σ, c𝔄/=[[c𝔄]].

  3. 3.

    For every natural numberMathworldPlanetmath n and every n-ary function symbol F of Σ,

  4. 4.

    For every natural number n and every n-ary relation symbol R of Σ, R𝔄/([[a1]],,[[an]]) if and only if for some aiai we have R𝔄(a1,,an).

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