Dergiler / Turkish Journal of Mathematics / 2015 / Cilt: 39 - Sayı: 5
On the second homology of the Sch\"{u}tzenberger product of monoids
- Sayfa
- 763–772
- DOI
- —
Özet
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 $T$.
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 $T$.