Editing Isomorphism theorems (section)

== Universal algebra ==
To generalise this to [[universal algebra]], normal subgroups need to be replaced by [[congruence relation]]s.

A '''congruence''' on an [[universal algebra|algebra]] <math>A</math> is an [[equivalence relation]] <math>\Phi\subseteq A \times A</math> that forms a subalgebra of <math>A \times A</math> considered as an algebra with componentwise operations.  One can make the set of [[equivalence class]]es <math>A/\Phi</math> into an algebra of the same type by defining the operations via representatives; this will be [[well-defined]] since <math>\Phi</math> is a subalgebra of <math>A \times A</math>. The resulting structure is the [[quotient (universal algebra)|quotient algebra]].

=== Theorem A (universal algebra)===
Let <math>f:A \rightarrow B</math> be an algebra [[homomorphism]].  Then the image of <math>f</math> is a subalgebra of <math>B</math>, the relation given by <math>\Phi:f(x)=f(y)</math> (i.e. the [[Kernel (set theory)|kernel]] of <math>f</math>) is a congruence on <math>A</math>, and the algebras <math>A/\Phi</math> and <math>\operatorname{im} f</math> are [[isomorphic]]. (Note that in the case of a group, <math>f(x)=f(y)</math> [[iff]] <math>f(xy^{-1}) = 1</math>, so one recovers the notion of kernel used in group theory in this case.)

=== Theorem B (universal algebra)===
Given an algebra <math>A</math>, a subalgebra <math>B</math> of <math>A</math>, and a congruence <math>\Phi</math> on <math>A</math>, let <math>\Phi_B = \Phi \cap (B \times B)</math> be the trace of <math>\Phi</math> in <math>B</math> and <math>[B]^\Phi=\{K \in A/\Phi: K \cap B \neq\emptyset\}</math> the collection of equivalence classes that intersect <math>B</math>. Then
# <math>\Phi_B</math> is a congruence on <math>B</math>,
# <math> \ [B]^\Phi</math> is a subalgebra of <math>A/\Phi</math>, and
# the algebra <math>[B]^\Phi</math> is isomorphic to the algebra <math>B/\Phi_B</math>.

=== Theorem C (universal algebra) ===
Let <math>A</math> be an algebra and <math>\Phi, \Psi</math> two congruence relations on <math>A</math> such that <math>\Psi \subseteq \Phi</math>. Then <math>\Phi/\Psi = \{ ([a']_\Psi,[a'']_\Psi): (a',a'')\in \Phi\} = [\ ]_\Psi \circ \Phi \circ [\ ]_\Psi^{-1}</math> is a congruence on <math>A/\Psi</math>, and <math>A/\Phi</math> is isomorphic to <math>(A/\Psi)/(\Phi/\Psi).</math>

=== Theorem D (universal algebra) ===
Let <math>A</math> be an algebra and denote <math>\operatorname{Con}A</math> the set of all congruences on <math>A</math>. The set
<math>\operatorname{Con}A</math> is a [[complete lattice]] ordered by inclusion.<ref>Burris and Sankappanavar (2012), p. 37</ref>
If <math>\Phi\in\operatorname{Con}A</math> is a congruence and we denote by <math>\left[\Phi,A\times A\right]\subseteq\operatorname{Con}A</math> the set of all congruences that contain <math>\Phi</math> (i.e. <math>\left[\Phi,A\times A\right]</math> is a principal [[filter (mathematics)|filter]] in <math>\operatorname{Con}A</math>, moreover it is a sublattice), then
the map  <math>\alpha:\left[\Phi,A\times A\right]\to\operatorname{Con}(A/\Phi),\Psi\mapsto\Psi/\Phi</math> is a lattice isomorphism.<ref>Burris and Sankappanavar (2012), p. 49</ref><ref>{{cite web |first1=William |last1=Sun |title=Is there a general form of the correspondence theorem? |url=https://math.stackexchange.com/q/2850331 |website=Mathematics StackExchange |access-date=20 July 2019}}</ref>