theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only. Not all algebraic structures in a wider sense fall into this scope. For example, ordered groups involve an ordering relation, so would not fall within this scope. The class of fields is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot be added to the type). One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products. For example, a topological group is just a group in the category of topological spaces. Examples Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities. Groups As an example, consider the definition of a group. Usually a group is defined in terms of a single binary operation ∗, subject to the axioms: Associativity (as in the previous section): x ∗ (y ∗ z) = (x ∗ y) ∗ z; formally: ∀x,y,z. x∗(y∗z)=(x∗y)∗z. Identity element: There exists an element e such that for each element x, one has e ∗ x = x = x ∗ e; formally: ∃e ∀x. e∗x=x=x∗e. Inverse element: The identity element is easily seen to be unique, and is usually denoted by e. Then for each x, there exists an element i such that x ∗ i = e = i ∗ x; formally: ∀x ∃i. x∗i=e=i∗x. (Some authors also use the "closure" axiom that x ∗ y belongs to A whenever x and y do, but here this is already implied by calling ∗ a binary operation.) This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation e and a unary