For a monoid M of hypersubstitutions, the collection of all M-solid varieties forms a complete sublattice of the lattice L(τ) of all varieties of a given type τ. Therefore, by the study of monoids of hypersubstitutions one can get more insight into the structure of the lattice L(τ). In particular, monoids of hypersubstitutions were studied in [9] as well as in [5]. We will give a complete characterization of all maximal submonoids of the monoid Reg(n) of all regular hypersubstitutions of type τ = (n) (introduced in [4]). The concept of a transformation hypersubstitution, introduced in [1], gives a relationship between monoids of hypersubstitutions and transformation semigroups. In the present paper, we apply the recent results about transformation semigroups by I. Guydzenov and I. Dimitrova ([11], [12]) to describe monoids of transformation hypersubstitutions.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1120,
author = {Ilinka Dimitrova and J\"org Koppitz},
title = {Maximal submonoids of monoids of hypersubstitutions},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {26},
year = {2006},
pages = {69-85},
zbl = {1136.08005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1120}
}
Ilinka Dimitrova; Jörg Koppitz. Maximal submonoids of monoids of hypersubstitutions. Discussiones Mathematicae - General Algebra and Applications, Tome 26 (2006) pp. 69-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1120/
[000] [1] V. Budd, K. Denecke and S.L. Wismath, Short Solid Superassociative Type (n) Varieties, East-West Journal of Mathematics 2 (2) (2001), 129-145. | Zbl 1008.08004
[001] [2] Th. Changphas, Monoids of Hypersubstitutions, Dissertation, Universität Potsdam 2004. | Zbl 1108.08300
[002] [3] Th. Changphas and K. Denecke, Full Hypersubstitutions and Full Solid Varieties of Semigroups, East-West Journal of Mathematics 4 (1) (2002), 177-193.
[003] [4] K. Denecke and J. Koppitz, M-Solid Varieties of Semigroups, Discuss. Math. 15 (1995), 23-41. | Zbl 0842.20050
[004] [5] K. Denecke and J. Koppitz, M-Solid Varieties of Algebras, Springer Science-Business Media 2006. | Zbl 1094.08001
[005] [6] K. Denecke, J. Koppitz and S. Niwczyk, Equational theories generated by hypersubstitutions of type] (n), Int. Journal of Algebra and Computation 12 (6) (2002), 867-876. | Zbl 1051.08003
[006] [7] K. Denecke, J. Koppitz and S. Shtrakov, The Depth of a Hypersubstitution, Journal of Automata, Languages and Combinatorics 6 (3) (2001), 253-262. | Zbl 0993.68052
[007] [8] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9, Wien 1995, 117-126. | Zbl 0884.08008
[008] [9] K. Denecke and S.L. Wismath, Hyperidentities and clones, Gordon and Breach Scientific Publishers, 2000.
[009] [10] K. Denecke and S.L. Wismath, Complexity of Terms, Composition, and Hypersubstitution, Int. Journal of Mathematics and Mathematical Sciences 15 (2003), 959-969. | Zbl 1015.08005
[010] [11] V.H. Fernandes, G.M.S. Gomes and M.M. Jesus, Presentations for Some Monoids of Partial Transformations on a Finite Chain, Communications in Algebra 33 (2005), 587-604. | Zbl 1072.20079
[011] [12] Il. Gyudzhenov and Il. Dimitrova, On the Maximal Subsemigroups of the Semigroup of All Isotone Transformations with Defect ≥ 2, Comptes rendus de l'Academie bulgare des Sciences 59 (3) (2006), 239-244. | Zbl 1104.20059
[012] [13] Il. Gyudzhenov and Il. Dimitrova, On the Maximal Subsemigroups of the Semigroup of all Monotone Transformations, Discuss. Math., submitted.
[013] [14] J.M. Howie, An Introduction to Semigroup Theory, Academic Press, London 1976. | Zbl 0355.20056
[014] [15] M.W. Liebeck, C.E. Praeger and J. Saxl, A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups, Journal of Algebra 111 (1987), 365-383. | Zbl 0632.20011
[015] [16] X. Yang, A Classification of Maximal Subsemigroups of Finite Order-Preserving Transformation Semigroups, Communications in Algebra 28 (3) (2000), 1503-1513. | Zbl 0948.20039