Common information (CI) as given by Gacs et al has played a major role in
network information theory. It is a measure of degeneracy in the joint PMF of
a
pair of random variables and is characterized using a pair of univariate
functions. Constructing the pair of random variables, that characterize
messages
recovered by two receiver terminals, with non-trivial CI enlarges the
achievable
rate region of interference channels and broadcast channels. We present a
generalization of CI to three random variables based on both univariate and
bivariate functions of the triple, and is characterized as a seven-dimensional
vector. We connect these bivariate functions with algebraic structures and
develop new coding theorems for a class of multi-terminal communication
problems.