Algebraic Representations of Entropy and Fixed-Parity Information Quantities

Many information-theoretic quantities have corresponding representations in terms of sets. The prevailing signed measure space for characterising entropy, the $I$-measure of Yeung, is occasionally unable to discern between qualitatively distinct systems. In previous work, we presented a refinement of this signed measure space and demonstrated its capability to represent many quantities, which we called logarithmically decomposable quantities. In the present work we demonstrate that this framework has natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper-sets), and we show that this behaviour allows us to make various counting arguments and characterise many fixed-parity information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables $X$, $Y$ and $Z = f(X,Y)$ is the XOR gate.