3. Terms and varieties

• A term $t$ or $t(x_1,\dots, x_n)$ of type $\tau$ is a formal expression as a string of symbols, defined recursively as follows, starting from variable symbols $x_1,\dots, x_n$ for $\tau$:

  (a) Each $x_i$ is a term, and

  (b) if $t_1,\dots, t_{n_\gamma}$ are terms, so is    f$_\gamma (t_1,\dots, t_{n_\gamma})$, where    f$_\gamma$ and the commas and parentheses are symbols and $\gamma \in \Gamma$.

  A term $t$ in variable symbols $x_1,\dots, x_n$ is often described by $t(x _ 1,\dots, x _ {n})$. For algebras with familiar notations we use those notations instead; for example, a group term might be written as $x _ 1 (x _ 2 ^ {-1} x _ {3})$.

• For elements $a_1,\dots, a_n$ of an algebra $A$ and term $t(x_1,\dots, x_n)$, the value $t(a_1,\dots, a_n)$ is the element of $A$ obtained by using the operations of $A$ while following $t(x_1,\dots, x_n)$ as a recipe.

  Thus $t$ induces a function on $A ^ n \rightarrow A$. The functions so induced are called the n-ary term functions on $A$. (For polynomial functions, see below.)

• A term relation $t_1 (a_1,\dots, a_n) = t_2 (a_1,\dots ,a_n)$ is an equation holding for a particular $n$-tuple of elements of $A$.

• A law is a formal equation $t_1 = t_2$ or $t_1 (x _ 1,\dots, x_n) = t_2 (x_1,\dots, x_n)$, with $(\forall x_1)\dots (\forall x_n)$ understood. Many authors write $t_1 \approx t_2$ to distinguish such formal laws from equations involving elements. The law $t_1 = t_2$ holds in $A$ when all $n$-tuples $a_1,\dots, a_n$ from $A$ satisfy the term relation $t_1 (a_1,\dots, a_n) = t_2 (a_1,\dots ,a_n)$.

  We also say $A$ satisfies $t_1 = t_2$, or $A$ is a model of $t_1 = t_2$, or write $A \models t_1=t_2$.

• A variety of algebras of a given type is the class of all models of some set of laws. Examples are the varieties of all groups, of all abelian groups, of all lattices, of all distributive lattices, and of other kinds of algebras whose laws are given in §.

  If $A$ is an algebra we write Var$(A)$ for the variety determined by all laws holding in $A$, which is the smallest variety containing $A$.

• A polynomial or polynomial function on $A$ is a term function in which some entries may be held constant. For example, if $A =$   R as a ring, $f(x) = 2x = x + x$ is a term function while $g(x) = \pi x$ is not a term function but is a polynomial function, obtained from the term function $h(x,y) = xy$ by $g(x) = h(\pi, x)$.