One can perform algebraic operations on power series to generate new power series.

In other words, z is an indeterminate variable allowing one to perform various algebraic operations among the jets.

The algebraic operations on are defined via these maps in the obvious manner.

Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations.

The basic algebraic operations for real interval numbers (real closed intervals) can be extended to complex numbers.

The ordering of integers is compatible with the algebraic operations in the following way:

The generalized selection is expressible with other basic algebraic operations.

A very important generalization principle used in fuzzification of algebraic operations is a closure property.

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions.

Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations.