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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1002,
author = {Klaus Denecke and Jennifer Hyndman and Shelly L. Wismath},
title = {The Galois correspondence between subvariety lattices and monoids of hpersubstitutions},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {20},
year = {2000},
pages = {21-36},
zbl = {0961.08006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1002}
}
Klaus Denecke; Jennifer Hyndman; Shelly L. Wismath. The Galois correspondence between subvariety lattices and monoids of hpersubstitutions. Discussiones Mathematicae - General Algebra and Applications, Tome 20 (2000) pp. 21-36. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1002/
[000] [1] J. Aczel, Proof of a theorem of distributive type hyperidentities, Algebra Universalis 1 (1971), 1-6. | Zbl 0219.08008
[001] [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). | Zbl 0135.03503
[002] [3] P.A. Birjukov, Varieties of idempotent semigroups (Russian), Algebra i Logika 9 (1970), 255-273.
[003] [4] K. Denecke, Pre-solid varieties, Demonstratio Math. 27 (1994), 741-750. | Zbl 0841.08006
[004] [5] K. Denecke and J. Koppitz, Hyperassociative varieties of semigroups, Semigroup Forum 49 (1994), 41-48. | Zbl 0806.20049
[005] [6] K. Denecke and J. Koppitz, Presolid varieties of semigroups, Arch. Math. (Brno) 31 (1995), 171-181. | Zbl 0842.20049
[006] [7] K. Denecke and J. Koppitz, M-solid varieties of semigroups, Discuss. Math.- Algebra and Stochastic Methods 15 (1995), 23-41. | Zbl 0842.20050
[007] [8] K. Denecke and J. Koppitz, Finite monoids of hypersubstitutions of type τ = (2), Semigroup Forum 56 (1998), 265-275. | Zbl 1080.20502
[008] [9] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes and clone congruences, Contributions to General Algebra 7 (1991), 97-118. | Zbl 0759.08005
[009] [10] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9 (1995), 117-126. | Zbl 0884.08008
[010] [11] K. Denecke and S.L. Wismath, Solid varieties of semigroups, Semigroup Forum 48 (1994), 219-234. | Zbl 0797.20045
[011] [12] K. Denecke and S.L. Wismath, Hyperidentities and Clones, Gordon & Breach Sci. Publ, London 2000. | Zbl 0960.08001
[012] [13] C. Fennemore, All varieties of bands. I and II, Math. Nachr. 48 (1971), 237-252 and 253-262. | Zbl 0194.02703
[013] [14] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195-224. | Zbl 0194.02701
[014] [15] J.A. Gerhard and M. Petrich, Varieties of bands revisited, Proc. London Math. Soc. (3) 58 (1989), 323-350. | Zbl 0676.20038
[015] [16] E. Graczyńska and D. Schweigert, Hypervarieties of a given type, Algebra Universalis 27 (1990), 305-318. | Zbl 0715.08002
[016] [17] J. P onka, Proper and inner hypersubstitutions of varieties, General Algebra and Ordered Sets, Palacký Univ., Olomouc 1994, 106-115.
[017] [18] L. Polák, On hyperassociativity, Algebra Universalis 36 (1996), 363-378.
[018] [19] D. Schweigert, Hyperidentities, Algebras and Orders, Kluwer Acad. Publ., Dordrecht 1993, 405-505.
[019] [20] W. Taylor, Hyperidentities and hypervarieties, Aequationes Math. 23 (1981), 111-127.
[020] [21] S.L. Wismath, Hyperidentities for some varieties of semigroups, Algebra Universalis 27 (1990), 111-127. | Zbl 0692.08008
[021] [22] S.L. Wismath, Hyperidentities for some varieties of commutative semigroups, Algebra Universalis 28 (1991), 245-273. | Zbl 0741.20043