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

Authors : MELEK YAĞCI, LEYLA BUGAY, HAYRULLAH AYIK

Pages : 763-772

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Publication Date : 0000-00-00

Article Type : Research Paper

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 $$H_{2}insert ignore into journalissuearticles values(S\Diamond T);=H_{2}insert ignore into journalissuearticles values(S);\times H_{2}insert ignore into journalissuearticles values(T);\times insert ignore into journalissuearticles values(H_{1}insert ignore into journalissuearticles values(S);\otimes _{\mathbb Z} H_{1}insert ignore into journalissuearticles values(T);); $$ 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