The ordered exponential (also called the path-ordered exponential) is a mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function, defined by means of a function from real numbers to a real or complex associative algebra. In practice the values lie in matrix and operator algebras.
For the element A(t) from the algebra (set g with the non-commutative product *), where t is the "time parameter", the ordered exponential of A can be defined via one of several equivalent approaches:
where the time moments are defined as for , and .