Presolid varieties of n-semigroups

Avapa Chantasartrassmee; Jörg Koppitz

Discussiones Mathematicae - General Algebra and Applications (2005)

  • Volume: 25, Issue: 2, page 221-233
  • ISSN: 1509-9415

Abstract

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he class of all M-solid varieties of a given type t forms a complete sublattice of the lattice ℒ(τ) of all varieties of algebrasof type t. This gives a tool for a better description of the lattice ℒ(τ) by characterization of complete sublattices. In particular, this was done for varieties of semigroups by L. Polák ([10]) as well as by Denecke and Koppitz ([4], [5]). Denecke and Hounnon characterized M-solid varieties of semirings ([3]) and M-solid varieties of groups were characterized by Koppitz ([9]). In the present paper we will do it for varieties of n-semigroups. An n-semigroup is an algebra of type (n), where the operation satisfies the [i,j]-associative laws for 1 ≤ i ≤ j ≤ n, introduced by Dörtnte ([2]). It is clear that the notion of a 2-semigroup is the same as the notion of a semigroup. Here we will consider the case n ≥ 3.

How to cite

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Avapa Chantasartrassmee, and Jörg Koppitz. "Presolid varieties of n-semigroups." Discussiones Mathematicae - General Algebra and Applications 25.2 (2005): 221-233. <http://eudml.org/doc/287688>.

@article{AvapaChantasartrassmee2005,
abstract = {he class of all M-solid varieties of a given type t forms a complete sublattice of the lattice ℒ(τ) of all varieties of algebrasof type t. This gives a tool for a better description of the lattice ℒ(τ) by characterization of complete sublattices. In particular, this was done for varieties of semigroups by L. Polák ([10]) as well as by Denecke and Koppitz ([4], [5]). Denecke and Hounnon characterized M-solid varieties of semirings ([3]) and M-solid varieties of groups were characterized by Koppitz ([9]). In the present paper we will do it for varieties of n-semigroups. An n-semigroup is an algebra of type (n), where the operation satisfies the [i,j]-associative laws for 1 ≤ i ≤ j ≤ n, introduced by Dörtnte ([2]). It is clear that the notion of a 2-semigroup is the same as the notion of a semigroup. Here we will consider the case n ≥ 3.},
author = {Avapa Chantasartrassmee, Jörg Koppitz},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {hypersubstitution; presolid; n-semigroup; -semigroup},
language = {eng},
number = {2},
pages = {221-233},
title = {Presolid varieties of n-semigroups},
url = {http://eudml.org/doc/287688},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Avapa Chantasartrassmee
AU - Jörg Koppitz
TI - Presolid varieties of n-semigroups
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2005
VL - 25
IS - 2
SP - 221
EP - 233
AB - he class of all M-solid varieties of a given type t forms a complete sublattice of the lattice ℒ(τ) of all varieties of algebrasof type t. This gives a tool for a better description of the lattice ℒ(τ) by characterization of complete sublattices. In particular, this was done for varieties of semigroups by L. Polák ([10]) as well as by Denecke and Koppitz ([4], [5]). Denecke and Hounnon characterized M-solid varieties of semirings ([3]) and M-solid varieties of groups were characterized by Koppitz ([9]). In the present paper we will do it for varieties of n-semigroups. An n-semigroup is an algebra of type (n), where the operation satisfies the [i,j]-associative laws for 1 ≤ i ≤ j ≤ n, introduced by Dörtnte ([2]). It is clear that the notion of a 2-semigroup is the same as the notion of a semigroup. Here we will consider the case n ≥ 3.
LA - eng
KW - hypersubstitution; presolid; n-semigroup; -semigroup
UR - http://eudml.org/doc/287688
ER -

References

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  1. [1] V. Budd, K. Denecke and S.L. Wismath, Short-solid superassociative type (n) varieties, East-West J. of Mathematics 3 (2) (2001), 129-145. Zbl1008.08004
  2. [2] W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z. 29 (1928), 1-19. Zbl54.0152.01
  3. [3] K. Denecke and Hounnon, All solid varieties of semirings, Journal of Algebra 248 (2002), 107-117. 
  4. [4] K. Denecke and J. Koppitz, Pre-solid varieties of semigroups, Archivum Mathematicum 31 (1995), 171-181. Zbl0842.20049
  5. [5] K. Denecke and J. Koppitz, Finite monoids of hypersubstitutions of type t = (2), Semigroup Forum 56 (1998), 265-275. Zbl1080.20502
  6. [6] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9 (1995), 117-126. Zbl0884.08008
  7. [7] K. Denecke, J. Koppitz and S.L. Wismath, Solid varieties of arbitrary type, Algebra Universalis 48 (2002), 357-378. Zbl1064.08006
  8. [8] K. Denecke and S.L. Wismath, Hyperidentities and clones, Gordon and Breach Scientific Publisher, 2000. 
  9. [9] J. Koppitz, Hypersubstitutions and groups, Novi Sad J. Math. 34 (2) (2004), 127-139. Zbl1212.20059
  10. [10] L. Polák, All solid varieties of semigroups, Journal of Algebra 219 (1999), 421-436. Zbl0935.20050
  11. [11] J. Płonka, Proper and inner hypersubstitutions of varieties, 'Proceedings of the International Conference: 'Summer School on General Algebra and Ordered Sets', Olomouc 1994', Palacký University, Olomouc 1994, 106-115. Zbl0828.08003

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