On the second homology of the Sch\"{u}tzenberger product of monoids
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 $T$.
Anahtar Kelimeler:
Monoid, Sch\"{u}tzenberger product, second integral homology, efficiency
On the second homology of the Sch\"{u}tzenberger product of monoids
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,
___
- Ayık H, Campbell CM, O’Connor JJ, Ruskuc N. Minimal presentations and efficiency of semigroups. Semigroup Forum 2000; 60: 231–242.
- Ayık H, Campbell CM, O’Connor JJ, Ruskuc N. On the efficiency of finite simple semigroups. Turk J Math 2000; 24: 129–146.
- Ayık H, Campbell CM, O’Connor JJ. On the efficiency of the direct products of Monogenic Monoids. Algebra
- Colloq 2007; 14: 279–284.
- Campbell CM, Mitchell JD, Ruskuc N. On defining groups efficiently without using inverses. Math P Camb Phil Soc 2002; 133: 31–36.
- C¸ evik AS. Minimal but inefficient presentations of the semi-direct products of some monoids. Semigroup Forum 2003; 66: 1–17.
- C¸ evik AS. The p -Cockcroft property of the semi-direct products of monoids. Int J Algebra Comput 2003; 13: 1–16.
- Guba VS, Pride SJ. Low dimensional (co)homology of free Burnside monoids. J Pure Appl Algebra 1996; 108: 61–79.
- Guba VS, Pride SJ. On the left and right cohomological dimension of monoids. Bull London Math Soc 1998; 30: 391–396.
- Howie JM. Fundamentals of Semigroup Theory. New York, NY, USA: Oxford University Press, 1995.
- Howie JM, Ruskuc N. Constructions and presentations for monoids. Comm Algebra 1994; 49: 6209–6224.
- Johnson DL. Presentations of Groups. Cambridge, UK: Cambridge University Press, 1990.
- Squier C. Word problems and a homological finiteness condition for monoids. J Pure Appl Algebra 1987; 49: 201–217.
- ISSN: 1300-0098
- Yayın Aralığı: Yılda 6 Sayı
- Yayıncı: TÜBİTAK
Sayıdaki Diğer Makaleler
$r$-ideals in commutative rings
Quadratic recursive towers of function fields over F2
Seher TUTDERE, Henning STICHTENOTH
On isophote curves and their characterizations
Defect polynomials and Tutte polynomials of some asymmetric graphs
Eunice MPHAKO-BANDA, Toufik MANSOUR
Quadratic recursive towers of function fields over $\mathbb{F}_2$
HENNING STICHTENOTH, SEHER TUTDERE
Super d-anti-magic labeling of subdivided $kC_{5}$
Generalized weakly central reduced rings
On the second homology of the Sch¨utzenberger product of monoids
Leyla BUĞAY, Hayrullah AYIK, Melek YAGCI
On the second homology of the Sch\"{u}tzenberger product of monoids