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

The minimal model program for b-log canonical divisors and applications