Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra "T". Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure "E". The quotient algebra "T"/"E" is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator "m", taking two arguments, and the inverse operator "i", taking one argument, and the identity element "e", a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables "x", "y", "z", etc. the term algebra is the collection of all possible terms involving "m", "i", "e" and the variables; so for example, "m(i(x), m(x,m(y,e)))" would be an element of the term algebra. One of the axioms defining a group is the identity "m(x, i(x)) = e"; another is "m(x,e) = x". The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either:   Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because formula_1, but fields do not have zero divisors. Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of "objects" with associated "morphisms." Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure.