On the second homology of the Sch\"{u}tzenberger product of monoids

Authors: MELEK YAĞCI, LEYLA BUGAY, HAYRULLAH AYIK

Abstract: For two finite monoids $S$ and $T$, we prove that the second integral homology of the Sch\"{u}tzenberger product $S\Diamond T$ is equal to $$H_{2}(S\Diamond T)=H_{2}(S)\times H_{2}(T)\times (H_{1}(S)\otimes _{\mathbb Z} H_{1}(T)) $$ as the second integral homology of the direct product of two monoids. Moreover, we show that $S\Diamond T$ is inefficient if there is no left or right invertible element in both $S$ and $T$.

Keywords: Monoid, Sch\"{u}tzenberger product, second integral homology, efficiency

Full Text: PDF