Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-31T00:14:20.486Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

Part of: Ordered sets Combinatorics Semigroups

Published online by Cambridge University Press:  06 December 2010

D. G. FITZGERALD  and
KWOK WAI LAU
D. G. FITZGERALD*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, TAS 7001, Australia (email: D.FitzGerald@utas.edu.au)
KWOK WAI LAU
Affiliation:
CSIRO Mathematics, Informatics and Statistics, Private Bag 5, Wembley, WA 6913, Australia (email: Rex.Lau@csiro.au)
*
✉For correspondence; e-mail: D.FitzGerald@utas.edu.au
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.

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

Keywords

partition monoidBrauer monoidideal structurenatural order

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 05A18: Partitions of sets 06A15: Galois correspondences, closure operators

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 273 - 288
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Borisavljević, M., Došen, K. and Petrić, Z., ‘Kauffman monoids’, J. Knot Theory Ramifications 11 (2002), 127–143.CrossRefGoogle Scholar
[2]Brauer, R. D., ‘On algebras which are connected with the semisimple continuous groups’, Ann. of Math. (2) 38 (1937), 857–872.CrossRefGoogle Scholar
[3]East, J., ‘Generators and relations for partition monoids and algebras’, Preprint, 2007.Google Scholar
[4]FitzGerald, D. G. and Leech, J., ‘Dual symmetric inverse monoids and representation theory’, J. Aust. Math. Soc. Ser. A 64 (1998), 345–367.CrossRefGoogle Scholar
[5]Green, J. A., ‘On the structure of semigroups’, Ann. of Math. (2) 54 (1951), 163–172.CrossRefGoogle Scholar
[6]Hall, T. E., ‘Congruences and Green’s relations on regular semigroups’, Glasg. Math. J. 13 (1972), 167–175.CrossRefGoogle Scholar
[7]Halverson, T. and Ram, A., ‘Partition algebras’, European J. Combin. 26 (2005), 869–921.CrossRefGoogle Scholar
[8]Howie, J. M., Fundamentals of Semigroup Theory (Oxford University Press, Oxford, 1995).Google Scholar
[9]Kudryavtseva, G. and Maltcev, V., ‘Two generalisations of the symmetric inverse semigroups’, Preprint, 2008, arXiv:math/0602623v2 [math.GR].Google Scholar
[10]Kudryavtseva, G., Maltcev, V. and Mazorchuk, V., ‘ℒ- and ℛ-cross-sections in the Brauer semigroup’, Semigroup Forum 72 (2006), 223–248.CrossRefGoogle Scholar
[11]Lau, K. W. and FitzGerald, D. G., ‘Ideal structure of the Kauffman and related monoids’, Comm. Algebra 34 (2006), 2617–2629.CrossRefGoogle Scholar
[12]Maltcev, V., ‘On a new approach to the dual symmetric inverse monoid ℐ*X’, Internat. J. Algebra Comput. 17 (2007), 567–591.Google Scholar
[13]Martin, P., ‘Temperley–Lieb algebras for nonplanar statistical mechanics—the partition algebra construction’, J. Knot Theory Ramifications 3 (1994), 51–82.CrossRefGoogle Scholar
[14]Martin, P., ‘The structure of the partition algebra’, J. Algebra 183 (1996), 319–358.Google Scholar
[15]Mazorchuk, V., ‘Endomorphisms of ℬn, and 𝒞n’, Comm. Algebra 30 (2002), 3489–3513.CrossRefGoogle Scholar
[16]Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Math. Soc. 97 (1986), 384–388.Google Scholar
[17]Nordahl, T. E. and Scheiblich, H. E., ‘Regular *-semigroups’, Semigroup Forum 16 (1978), 369–377.Google Scholar
[18]Sweedler, M. E., ‘Multiplication alteration by two-cocycles’, Illinois J. Math. 15 (1971), 302–323.CrossRefGoogle Scholar
[19]Vernitskii, A., ‘A generalization of symmetric inverse semigroups’, Semigroup Forum 75 (2007), 417–426.Google Scholar
[20]Wilcox, S., ‘Cellularity of diagram algebras as twisted semigroup algebras’, J. Algebra 309 (2007), 10–31.CrossRefGoogle Scholar