4. Some theorems

Familiar theorems from group theory all generalize, except that in groups we focus on normal subgroups, but for algebras in general we focus on congruence relations, a generalization of the coset decomposition of a normal subgroup. The reason is that in groups the whole coset decomposition is determined by knowing the block containing the identity element, while for algebras in general no one block determines the rest.

• The subalgebra of $A$ generated by $g_1,\dots, g_n$ is the set of elements of the form $t(g_1,\dots, g_n)$ for some term $t$ in $n$ variables.

• The image of a homomorphism is a subalgebra.

• The set Con$(A)$ of all congruence relations on $A$ is a lattice, the congruence lattice of $A$.

• For $\theta \in$   Con$(A)$, the set $A/\theta$ of blocks is an algebra, in an obvious way, of the same type as $A$.

• If $\phi: A\rightarrow B$ is a homomorphism, then the equivalence relation on $A$ induced by $\phi$ is a congruence relation, which we call $\ker \phi$, the kernel of $\phi$.

  Observe that if $\theta \in$   Con$(A)$ and $\eta: A \rightarrow A/\theta$ is the natural surjection, then $\ker \eta = \theta$.

• If $\phi: A\rightarrow B$ is a surjective homomorphism, then $B \cong A/\ker \phi$ (the first isomorphism theorem).

  Thus we have an ``internal description'' of all the homomorphic images of $A$, up to isomorphism.

• If $\phi: A\rightarrow B$ is a surjective homomorphism, then the congruence relations on $B$ correspond one-to-one to the congruence relations on $A$ that contain $\ker \phi$ (the correspondence theorem).

• For a direct product $P = \prod _ {\gamma \in \Gamma} A _ \gamma$, for each $\gamma \in \Gamma$ the coordinate projection $\pi _ \gamma: P \rightarrow A _ \gamma$ is a surjective homomorphism.