In this section we develop integer ambiguity algebra, a mathematical approach to handle integer ambiguities between different GNSS frequencies and introduce what we call the ambiguity-free linear combination. We first show the vector form of the wide-lane ambiguity for multi-frequency GNSS and then develop integer ambiguity algebra and show in detail the integer property of the ionosphere-free ambiguity for GPS and Galileo. We show that any GNSS ionosphere-free linear combination can be represented by an integer ambiguity without resolving wide-lane ambiguity. This opens up the possibility of forming an integer ambiguity of arbitrary wavelength, when combined with narrow-lane ambiguity. We introduce an elegant way to resolve wide-/narrow-lane ambiguities using the ambiguity-free linear combination that is consistent with what we term absolute code biases. The advantage of this approach is the consistent resolution of wide-lane ambiguities and calibration of wide-lane biases in an absolute sense, since the same ambiguity-free linear combination can be used to estimate absolute code biases, (see section on absolute code biases). Code biases can be defined in an absolute sense if one uses the IGS convention for estimated clock parameters that the net effect of code biases is zero for the ionosphere-free linear combination of P-code measurements, or so-called P3-clocks. They are still limited by the full number of wide-lane ambiguities that can be defined separately for two- and three-carriers with a wavelength of 0.67 m and 3.41 m respectively. Since absolute code biases are determined against the ionosphere-free P-code, we obtain a consistent framework for ambiguity resolution for all four GNSS. Then, by using integer ambiguity algebra, we develop three-carrier wide-/narrow-lane linear combinations for GPS/Galileo and show how to use this approach for ambiguity resolution and retrieval of ionospheric effects. We show that a three-carrier-type Melbourne-Wubbena linear combination can be derived by means of ambiguity algebra.