Isomorphism theorems

The isomorphism theorems are a set of four theorems, most generally statable in the language of universal algebra: the congruence relation and quotient. For particular examples, see

First theorem

Let $𝑓 : 𝐴 \to 𝐵$ be an algebra homomorphism. Then $i m 𝑓$ is a subalgebra of $𝐴$ , the relation $𝑥 \equiv 𝑦 ⟺ 𝑓 (𝑥) = 𝑓 (𝑦)$ is a congruence and $i m 𝑓$ and $𝐴 / ≅$ are isomorphic. #m/thm/algebra

Proof

#missing/proof

Second theorem

Let $𝐴$ be an algebra, $𝐵$ a subalgebra of $𝐴$ , and $\equiv$ be a congruence on $𝐴$ . Let further $\equiv 𝐵 = \equiv \cap (𝐵 \times 𝐵)$ be the restriction of $\equiv$ to $𝐵$ and

[𝐵] \equiv = {𝐾 \in 𝐴 / \equiv : 𝐾 \cap 𝐵 \neq \emptyset}

be the equivalence glasses under $\equiv$ intersecting $𝐵$ . Then #m/thm/algebra

$\equiv 𝐵$ is a congruence on $𝐵$
$[𝐵] \equiv$ is a subalgebra of $𝐴 / \equiv$ isomorphic to $𝐵 / \equiv 𝐵$

Proof

#missing/proof

Third theorem

Let $𝐴$ be an algebra and $\equiv, \equiv'$ be congruences on $𝐴$ such that $\equiv' \subseteq \equiv$ . Then

\equiv / \equiv' = {([𝑎] \equiv', [𝑏] \equiv') : (𝑎, 𝑏) \in \equiv} = [-] \equiv' \circ \equiv \circ [-] - 1 \equiv'

is a congruence on $𝐴 / \equiv'$ and $𝐴 / \equiv$ is isomorphic to $(𝐴 / \equiv') / (\equiv / \equiv')$ . #m/thm/algebra

Proof

#missing/proof

Fourth isomorphism theorem

Also called the correspondence theorem

#state/develop | #lang/en | #SemBr