Isomorphism theorems
Theorem D (universal algebra)

Let A {\displaystyle A} be an algebra and denote Con A {\displaystyle \operatorname {Con} A} the set of all congruences on A {\displaystyle A}. The set Con A {\displaystyle \operatorname {Con} A} is a complete lattice ordered by inclusion. If Φ ∈ Con A {\displaystyle \Phi \in \operatorname {Con} A} is a congruence and we denote by ⊆ Con A {\displaystyle \left\subseteq \operatorname {Con} A} the set of all congruences that contain Φ {\displaystyle \Phi } (i.e. {\displaystyle \left} is a principal filter in Con A {\displaystyle \operatorname {Con} A}, moreover it is a sublattice), thenthe map α: → Con (A / Φ), Ψ ↦ Ψ / Φ {\displaystyle \alpha :\left\to \operatorname {Con} (A/\Phi),\Psi \mapsto \Psi /\Phi } is a lattice isomorphism.
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