# Maximal submonoids of monoids of hypersubstitutions

• Volume: 27, Issue: 1, page 69-85
• ISSN: 1509-9415

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## Abstract

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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.

## How to cite

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Ilinka Dimitrova, and Jörg Koppitz. "Maximal submonoids of monoids of hypersubstitutions." Discussiones Mathematicae - General Algebra and Applications 27.1 (2006): 69-85. <http://eudml.org/doc/276921>.

abstract = {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.},
author = {Ilinka Dimitrova, Jörg Koppitz},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {regular hypersubstitutions; maximal monoids of hypersubstitutions; transformation semigroups; hypersubstitution; maximal submonoid; transformation semigroup},
language = {eng},
number = {1},
pages = {69-85},
title = {Maximal submonoids of monoids of hypersubstitutions},
url = {http://eudml.org/doc/276921},
volume = {27},
year = {2006},
}

TY - JOUR
AU - Jörg Koppitz
TI - Maximal submonoids of monoids of hypersubstitutions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 27
IS - 1
SP - 69
EP - 85
AB - 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.
LA - eng
KW - regular hypersubstitutions; maximal monoids of hypersubstitutions; transformation semigroups; hypersubstitution; maximal submonoid; transformation semigroup
UR - http://eudml.org/doc/276921
ER -

## References

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12. [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. Zbl1104.20059
13. [13] Il. Gyudzhenov and Il. Dimitrova, On the Maximal Subsemigroups of the Semigroup of all Monotone Transformations, Discuss. Math., submitted.
14. [14] J.M. Howie, An Introduction to Semigroup Theory, Academic Press, London 1976. Zbl0355.20056
15. [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. Zbl0632.20011
16. [16] X. Yang, A Classification of Maximal Subsemigroups of Finite Order-Preserving Transformation Semigroups, Communications in Algebra 28 (3) (2000), 1503-1513. Zbl0948.20039

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