On the second homology of the Sch¨utzenberger product of monoids

Abstract

For two finite monoids S and T , we prove that the second integral homology of the Sch¨utzenberger product S✸T is equal to H2(S✸T) = H2(S) × H2(T) × (H1(S) ⊗Z H1(T)) as the second integral homology of the direct product of two monoids. Moreover, we show that S✸T is inefficient if there is no left or right invertible element in both S and T .