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In this note we consider monoidal complexes and their associated algebras,called toric face rings. These rings generalize Stanley-Reisner rings andaffine monoid algebras. We compute initial ideals of the presentation ideal ofa toric face ring, and determine its graded Betti numbers. Our resultsgeneralize celebrated theorems of Hochster in combinatorial commutativealgebra.

Gr\"obner bases and Betti numbers of monoidal complexes