Presentations for (singular) partition monoids: a new approach. (English) Zbl 1499.20127
Summary: We give new, short proofs of the presentations for the partition monoid and its singular ideal originally given in the author’s 2011 papers in [J. Algebra 339, No. 1, 1–26 (2011; Zbl 1277.20069)] and [Int. J. Algebra Comput. 21, No. 1–2, 147–178 (2011; Zbl 1229.20066)].
MSC:
| 20M05 | Free semigroups, generators and relations, word problems |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
References:
| [1] | Auinger, K.; Chen, Y.; Hu, X.; Luo, Y.; Volkov, M. V., The finite basis problem for Kauffman monoids, Algebra Universalis, 74, 333-350, (2015) · Zbl 1332.20056 · doi:10.1007/s00012-015-0356-x |
| [2] | Auinger, K., Pseudovarieties generated by Brauer type monoids, Forum Math., 26, 1-24, (2014) · Zbl 1305.20066 · doi:10.1515/form.2011.146 |
| [3] | Auinger, K.; Dolinka, I.; Volkov, M. V., Equational theories of semigroups with involution, J. Algebra, 369, 203-225, (2012) · Zbl 1294.20072 · doi:10.1016/j.jalgebra.2012.06.021 |
| [4] | Dolinka, I.; East, J.; Gray, R. D., Motzkin monoids and partial Brauer monoids, J. Algebra, 471, 251-298, (2017) · Zbl 1406.20065 · doi:10.1016/j.jalgebra.2016.09.018 |
| [5] | East, J., A presentation of the singular part of the symmetric inverse monoid, Comm. Algebra, 34, 1671-1689, (2006) · Zbl 1099.20029 · doi:10.1080/00927870500542689 |
| [6] | East, J., A presentation for the singular part of the full transformation semigroup, Semigroup Forum, 81, 357-379, (2010) · Zbl 1207.20058 · doi:10.1007/s00233-010-9250-1 |
| [7] | East, J., Generators and relations for partition monoids and algebras, J. Algebra, 339, 1-26, (2011) · Zbl 1277.20069 · doi:10.1016/j.jalgebra.2011.04.008 |
| [8] | East, J., On the singular part of the partition monoid, Internat. J. Algebra Comput., 21, 147-178, (2011) · Zbl 1229.20066 · doi:10.1142/S021819671100611X |
| [9] | East, J., A symmetrical presentation for the singular part of the symmetric inverse monoid, Algebra Universalis, 74, 207-228, (2015) · Zbl 1332.20061 · doi:10.1007/s00012-015-0347-y |
| [10] | East, J.; Fitzgerald, D. G., The semigroup generated by the idempotents of a partition monoid, J. Algebra, 372, 108-133, (2012) · Zbl 1281.20077 · doi:10.1016/j.jalgebra.2012.09.014 |
| [11] | East, J.; Gray, R. D., Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, J. Combin. Theory Ser. A, 146, 63-128, (2017) · Zbl 1351.05227 · doi:10.1016/j.jcta.2016.09.001 |
| [12] | Fitzgerald, D. G., A presentation for the monoid of uniform block permutations, Bull. Austral. Math. Soc., 68, 317-324, (2003) · Zbl 1043.20037 · doi:10.1017/S0004972700037692 |
| [13] | Fitzgerald, D. G.; Lau, K. W., On the partition monoid and some related semigroups, Bull. Aust. Math. Soc., 83, 273-288, (2011) · Zbl 1230.20061 |
| [14] | Fitzgerald, D. G.; Leech, J., Dual symmetric inverse monoids and representation theory, J. Austral. Math. Soc. Ser. A, 64, 345-367, (1998) · Zbl 0927.20040 · doi:10.1017/S1446788700039227 |
| [15] | Halverson, T.; Ram, A., Partition algebras, European J. Combin., 26, 869-921, (2005) · Zbl 1112.20010 · doi:10.1016/j.ejc.2004.06.005 |
| [16] | Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc., 41, 707-716, (1966) · Zbl 0146.02903 · doi:10.1112/jlms/s1-41.1.707 |
| [17] | Jones, V. F. R., Subfactors, The Potts model and the symmetric group, 259-267, (1994), World Sci. Publ.: World Sci. Publ., River Edge, NJ · Zbl 0938.20505 |
| [18] | Koenig, S., Trends in representation theory of algebras and related topics, A panorama of diagram algebras, 491-540, (2008), Zürich · Zbl 1210.16018 · doi:10.4171/062-1/12 |
| [19] | Lau, K. W.; Fitzgerald, D. G., Ideal structure of the Kauffman and related monoids, Comm. Algebra, 34, 2617-2629, (2006) · Zbl 1103.47032 · doi:10.1080/00927870600651414 |
| [20] | Lipscomb, S., Symmetric inverse semigroups, (1996), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 0857.20047 · doi:10.21236/ADA312447 |
| [21] | Maltcev, V.; Mazorchuk, V., Presentation of the singular part of the Brauer monoid, Math. Bohem., 132, 297-323, (2007) · Zbl 1163.20035 |
| [22] | Martin, P., Temperley-Lieb algebras for nonplanar statistical mechanics–the partition algebra construction, J. Knot Theory Ramifications, 3, 51-82, (1994) · Zbl 0804.16002 · doi:10.1142/S0218216594000071 |
| [23] | Martin, P., Noncommutative rings, group rings, diagram algebras and their applications. Contemp. Math., On diagram categories, representation theory and statistical mechanics, 99-136, (2008), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 1139.16002 · doi:10.1090/conm/456/08886 |
| [24] | Moore, E. H., Concerning the abstract groups of order k! and k!/2 holohedrically isomorphic with the symmetric and the alternating substitution-groups on k letters, Proc. London Math. Soc., 28, 357-366, (1897) · JFM 28.0121.03 |
| [25] | Popova, L. M., Defining relations in some semigroups of partial transformations of a finite set (in Russian), Uchenye Zap. Leningrad Gos. Ped. Inst., 218, 191-212, (1961) |
| [26] | Wilcox, S., Cellularity of diagram algebras as twisted semigroup algebras, J. Algebra, 309, 10-31, (2007) · Zbl 1154.16021 · doi:10.1016/j.jalgebra.2006.10.016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
