A definition of thermodynamic entropy can be based entirely on certain properties of the relation ≺ ≺ of adiabatic accessibility that are taken as axioms in the Lieb-Yngvason approach. In the following list of properties of the ≺ ≺ operator, a system is represented by a capital letter, e.g. X, Y or Z. A system X whose extensive parameters are multiplied by λ λ is written λ λ X . (e.g. for a simple gas, this would mean twice the amount of gas in twice the volume, at the same pressure.) A system consisting of two subsystems X and Y is written (X,Y). If X ≺ ≺ Y and Y ≺ ≺ X are both true, then each system can access the other and the transformation taking one into the other is reversible. This is an equivalence relationship written X ∼ ∼ A Y . Otherwise, it is irreversible. Adiabatic accessibility has the following properties: