  We discuss the minimal model program for b-log varieties, which is a pair of
a variety and a b-divisor, as a natural generalization of the minimal model
program for ordinary log varieties. We show that the main theorems of the log
MMP work in the setting of the b-log MMP. If we assume that the log MMP
terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes
both the log MMP and the equivariant MMP as special cases. There are various
interesting b-log varieties arising from different objects, including the
Brauer pairs, or "non-commutative algebraic varieties which are finite over
their centres". The case of toric Brauer pairs is discussed in further detail.
