The Galois correspondence between subvariety lattices and monoids of hpersubstitutions

Klaus Denecke; Jennifer Hyndman; Shelly L. Wismath

Discussiones Mathematicae - General Algebra and Applications (2000)

  • Volume: 20, Issue: 1, page 21-36
  • ISSN: 1509-9415

Abstract

top
Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.

How to cite

top

Klaus Denecke, Jennifer Hyndman, and Shelly L. Wismath. "The Galois correspondence between subvariety lattices and monoids of hpersubstitutions." Discussiones Mathematicae - General Algebra and Applications 20.1 (2000): 21-36. <http://eudml.org/doc/287704>.

@article{KlausDenecke2000,
abstract = {Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.},
author = {Klaus Denecke, Jennifer Hyndman, Shelly L. Wismath},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {hypersubstitutions; hyperidentities; M-hyperidentities; monoids of hypersubstitutions; varieties; solid varieties; M-solid varieties of bands; Galois correspondence; monoid; lattice of varieties of bands; complete sublattices},
language = {eng},
number = {1},
pages = {21-36},
title = {The Galois correspondence between subvariety lattices and monoids of hpersubstitutions},
url = {http://eudml.org/doc/287704},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Klaus Denecke
AU - Jennifer Hyndman
AU - Shelly L. Wismath
TI - The Galois correspondence between subvariety lattices and monoids of hpersubstitutions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2000
VL - 20
IS - 1
SP - 21
EP - 36
AB - Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.
LA - eng
KW - hypersubstitutions; hyperidentities; M-hyperidentities; monoids of hypersubstitutions; varieties; solid varieties; M-solid varieties of bands; Galois correspondence; monoid; lattice of varieties of bands; complete sublattices
UR - http://eudml.org/doc/287704
ER -

References

top
  1. [1] J. Aczel, Proof of a theorem of distributive type hyperidentities, Algebra Universalis 1 (1971), 1-6. Zbl0219.08008
  2. [2] V.D. Belousov, Systems of quasigroups with generalized identities, (Russian) Uspekhi Mat. Nauk. 20 (1965) 75-146 (English translation: Russian Math. Surveys 20 (1965) 75-143). Zbl0135.03503
  3. [3] P.A. Birjukov, Varieties of idempotent semigroups (Russian), Algebra i Logika 9 (1970), 255-273. 
  4. [4] K. Denecke, Pre-solid varieties, Demonstratio Math. 27 (1994), 741-750. Zbl0841.08006
  5. [5] K. Denecke and J. Koppitz, Hyperassociative varieties of semigroups, Semigroup Forum 49 (1994), 41-48. Zbl0806.20049
  6. [6] K. Denecke and J. Koppitz, Presolid varieties of semigroups, Arch. Math. (Brno) 31 (1995), 171-181. Zbl0842.20049
  7. [7] K. Denecke and J. Koppitz, M-solid varieties of semigroups, Discuss. Math.- Algebra and Stochastic Methods 15 (1995), 23-41. Zbl0842.20050
  8. [8] K. Denecke and J. Koppitz, Finite monoids of hypersubstitutions of type τ = (2), Semigroup Forum 56 (1998), 265-275. Zbl1080.20502
  9. [9] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes and clone congruences, Contributions to General Algebra 7 (1991), 97-118. Zbl0759.08005
  10. [10] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9 (1995), 117-126. Zbl0884.08008
  11. [11] K. Denecke and S.L. Wismath, Solid varieties of semigroups, Semigroup Forum 48 (1994), 219-234. Zbl0797.20045
  12. [12] K. Denecke and S.L. Wismath, Hyperidentities and Clones, Gordon & Breach Sci. Publ, London 2000. Zbl0960.08001
  13. [13] C. Fennemore, All varieties of bands. I and II, Math. Nachr. 48 (1971), 237-252 and 253-262. Zbl0194.02703
  14. [14] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195-224. Zbl0194.02701
  15. [15] J.A. Gerhard and M. Petrich, Varieties of bands revisited, Proc. London Math. Soc. (3) 58 (1989), 323-350. Zbl0676.20038
  16. [16] E. Graczyńska and D. Schweigert, Hypervarieties of a given type, Algebra Universalis 27 (1990), 305-318. Zbl0715.08002
  17. [17] J. P onka, Proper and inner hypersubstitutions of varieties, General Algebra and Ordered Sets, Palacký Univ., Olomouc 1994, 106-115. 
  18. [18] L. Polák, On hyperassociativity, Algebra Universalis 36 (1996), 363-378. 
  19. [19] D. Schweigert, Hyperidentities, Algebras and Orders, Kluwer Acad. Publ., Dordrecht 1993, 405-505. 
  20. [20] W. Taylor, Hyperidentities and hypervarieties, Aequationes Math. 23 (1981), 111-127. 
  21. [21] S.L. Wismath, Hyperidentities for some varieties of semigroups, Algebra Universalis 27 (1990), 111-127. Zbl0692.08008
  22. [22] S.L. Wismath, Hyperidentities for some varieties of commutative semigroups, Algebra Universalis 28 (1991), 245-273. Zbl0741.20043

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.