Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T05:13:28.315Z Has data issue: false hasContentIssue false

THE IDEMPOTENT-GENERATED SUBSEMIGROUP OF THE KAUFFMAN MONOID

Published online by Cambridge University Press:  01 March 2017

IGOR DOLINKA
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21101 Novi Sad, Serbia e-mail: [email protected]
JAMES EAST
Affiliation:
Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Auinger, K., Chen, Y., Hu, X., Luo, Y. and Volkov, M. V., The finite basis problem for Kauffman monoids, Algebra Universalis 74 (3–4) (2015), 333350.Google Scholar
2. Bokut, L. A.' and Li, D. V., The Gröbner-Shirshov basis for the Temperley-Lieb-Kauffman monoid, Izv. Ural. Gos. Univ. Mat. Mekh. 7 (36) (2005), 4966, 190.Google Scholar
3. Borisavljević, M., Došen, K. and Petrić, Z., Kauffman monoids, J. Knot Theory Ramifications 11 (2) (2002), 127143.Google Scholar
4. Dolinka, I. and East, J., Twisted Brauer monoids, to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics, arXiv:1510.08666.Google Scholar
5. Dolinka, I., East, J., Evangelou, A., FitzGerald, D., Ham, N., Hyde, J. and Loughlin, N., Idempotent statistics of the Motzkin, Jones and Kauffman monoids, Preprint, 2015, arXiv:1507.04838.Google Scholar
6. Dolinka, I., East, J. and Gray, R. D., Motzkin monoids and partial Brauer monoids, Journal of Algebra 471 (2017), 251298.Google Scholar
7. East, J., On the singular part of the partition monoid, Internat. J. Algebra Comput. 21 (1–2) (2011), 147178.Google Scholar
8. East, J. and Gray, R. D., Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals, J. Combin. Theory Ser. A, 146 (2017), 63128.Google Scholar
9. Erdos, J. A., On products of idempotent matrices, Glasgow Math. J. 8 (1967), 118122.CrossRefGoogle Scholar
10. Gray, R. D., The minimal number of generators of a finite semigroup, Semigroup Forum 89 (1) (2014), 135154.Google Scholar
11. Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707716.CrossRefGoogle Scholar
12. Howie, J. M., Idempotent generators in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4) (1978), 317323.Google Scholar
13. Howie, J. M., Idempotents in completely 0-simple semigroups, Glasgow Math. J. 19 (2) (1978), 109113.Google Scholar
14. Howie, J. M., Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12 (The Clarendon Press, Oxford University Press, New York, 1995, Oxford Science Publications).Google Scholar
15. Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1) (1983), 125.Google Scholar
16. Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (2) (1990), 417471.Google Scholar
17. Lau, K. W. and FitzGerald, D. G., Ideal structure of the Kauffman and related monoids, Comm. Algebra 34 (7) (2006), 26172629.Google Scholar
18. Maltcev, V. and Mazorchuk, V., Presentation of the singular part of the Brauer monoid, Math. Bohem. 132 (3) (2007), 297323.Google Scholar
19. Temperley, H. N. V. and Lieb, E. H., Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1549) (1971), 251280.Google Scholar
20. Wilcox, S., Cellularity of diagram algebras as twisted semigroup algebras, J. Algebra 309 (1) (2007), 1031.Google Scholar