2. Algebras and basic constructions

• A function $f: A ^ n \rightarrow A$ is an $n$-ary operation on $A$; $n$ is its ``arity.''

  (For $n=0,1,2,3$ we say ``nullary'', ``unary'', ``binary'', ``ternary''.)

• An algebra is a set $A$ with a given family of operations $f _ {\gamma}$ ( $\gamma \in \Gamma$), called the ``basic operations'' of $A$. Officially, the algebra is $\langle A; f _ {\gamma}, \gamma \in \Gamma \rangle$. Texts often use a separate letter to distinguish the algebra from the set, but we'll follow the informal practice of group theory and use $A$ for both.

• The type of $\langle A; f _ {\gamma}, \gamma \in \Gamma \rangle$ is the function $\tau: \Gamma \rightarrow \omega$ given by $\tau( \gamma ) = n _ {\gamma}$, the arity of $f _ {\gamma}$. Two algebras of the same type are similar. In discussions involving more than one algebra, we'll normally assume that all the algebras are similar. Usually $\Gamma$ will be finite; if $\vert\Gamma\vert = m$, then it is simplest to choose $\Gamma = {0,\dots, m-1}$ and write the $n _ {\gamma}$ as a sequence.

  For example, the type of a Boolean algebra $\langle B; \vee , \wedge , 0, 1, ' \rangle$ can be written $\langle 2,2,0,0,1 \rangle$.

• A subalgebra of an algebra $A$ is a subset $S \subseteq A$ that is closed under all the basic operations of $A$.

• An algebra $A$ is said to be generated by its elements $g_1,\dots, g_n$ if the smallest subalgebra of $A$ that contains all the $g_i$ is $A$ itself.

• A homomorphism $\phi: A\rightarrow B$ between similar algebras is a map compatible with the basic operations of $A$ and $B$.

• The direct product of a family of similar algebras, $A _ 1 \times A _ 2$ or more generally $\prod _ {\gamma \in \Gamma} A _ \gamma$, is the set-theoretic cartesian product with operations computed coordinatewise.

• A congruence relation on $A$ is an equivalence relation $\theta$ on $A$ that is compatible with the basic operations of $A$.

• For a congruence relation $\theta$ on $A$, the blocks of $\theta$ form an algebra $A/\theta$ of the same type, with a natural surjective homomorphism $\eta: A \rightarrow A/\theta$.