Abstract
Let I_n, S_n and A_n be the symmetric inverse semigroup, the symmetric group and the alternating group on X_n={1,…,n}, for n≥2, respectively. Also let I_(n,r) be the subsemigroup consists of all partial injective maps with height less than or equal to r for 1≤r≤n-1, and let SI_(n,r)=I_(n,r)∪S_n and AI_(n,r)=I_(n,r)∪A_n. A non-idempotent element whose square is an idempotent is called a quasi-idempotent. In this paper we obtain the rank and the quasi-idempotent rank of SI_(n,r) (of AI_(n,r)). Also we obtain the relative rank and the relative quasi-idempotent rank of SI_(n,r) modulo S_n (of AI_(n,r) modulo A_n).
References
- Ayık, G., Ayık, H., Howie, J. M., On factorisations and generators in transformation semigroup, Semigroup Forum, 70, 225–237, (2005). Ayık, G., Ayık, H., Howie, J. M., Ünlü, Y., Rank properties of the semigroup of singular transformations on a finite set, Communications in Algebra, 36, 2581–2587, (2008). Bugay L. Quasi-idempotent ranks of some permutation groups and transformation semigroups, Turkish Journal of Mathematics 43, 2390–2395, (2019). Ganyushkin, O., Mazorchuk, V., Classical Finite Transformation Semigroups, London, Springer-Verlag, (2009). Garba, G.U., On the idempotent ranks of certain semigroups of order-preserving transformations, Portugaliae Mathematica, 51, 185–204, (1994). Garba, G. U., Imam, A. T., Products of quasi-idempotents in finite symmetric inverse semigroups, Semigroup Forum, 92, 645–658, (2016). Howie, J. M., Fundamentals of Semigroup Theory. New York, Oxford University Press, (1995). Levi, I., McFadden, R. B., S_n-Normal semigroups, Proceedings of the Edinburgh Mathematical Society, 37, 471–476, (1994). Yiğit, E., Ayık, G., Ayık, H., Minimal relative generating sets of some partial transformation semigroups, Communications in Algebra, 45, 1239–1245, (2017). Zhao, P., Fernandes, V. H., The ranks of ideals in various transformation monoids, Communications in Algebra, 43, 674-692, (2015).
Abstract
n≥2 için I_n, S_n ve A_n, sırasıyla, X_n={1,…,n} üzerindeki simetrik inverse yarıgrup, simetrik grup ve alterne grup olsun. Ayrıca, 1≤r≤n-1 için I_(n,r), yüksekliği en fazla r olan tüm kısmi bire-bir dönüşümlerden oluşan altyarıgrup, SI_(n,r)=I_(n,r)∪S_n ve AI_(n,r)=I_(n,r)∪A_n olsun. Karesi idempotent olan fakat kendisi idempotent olmayan bir elemana quasi-idempotent denir. Bu calışmada SI_(n,r) (AI_(n,r)) nin rankını elde ettik. Ayrıca, modulo S_n e göre SI_(n,r) nin (modulo A_n e göre AI_(n,r) nin) ilişkili rankını ve quasi-ilişkili rankını elde ettik.
References
- Ayık, G., Ayık, H., Howie, J. M., On factorisations and generators in transformation semigroup, Semigroup Forum, 70, 225–237, (2005). Ayık, G., Ayık, H., Howie, J. M., Ünlü, Y., Rank properties of the semigroup of singular transformations on a finite set, Communications in Algebra, 36, 2581–2587, (2008). Bugay L. Quasi-idempotent ranks of some permutation groups and transformation semigroups, Turkish Journal of Mathematics 43, 2390–2395, (2019). Ganyushkin, O., Mazorchuk, V., Classical Finite Transformation Semigroups, London, Springer-Verlag, (2009). Garba, G.U., On the idempotent ranks of certain semigroups of order-preserving transformations, Portugaliae Mathematica, 51, 185–204, (1994). Garba, G. U., Imam, A. T., Products of quasi-idempotents in finite symmetric inverse semigroups, Semigroup Forum, 92, 645–658, (2016). Howie, J. M., Fundamentals of Semigroup Theory. New York, Oxford University Press, (1995). Levi, I., McFadden, R. B., S_n-Normal semigroups, Proceedings of the Edinburgh Mathematical Society, 37, 471–476, (1994). Yiğit, E., Ayık, G., Ayık, H., Minimal relative generating sets of some partial transformation semigroups, Communications in Algebra, 45, 1239–1245, (2017). Zhao, P., Fernandes, V. H., The ranks of ideals in various transformation monoids, Communications in Algebra, 43, 674-692, (2015).
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | April 10, 2020 |
| Submission Date | February 3, 2020 |
| Published in Issue | Year 2020 Volume: 22 Issue: 2 |