Medial modes and rectangular algebras

Anna Zamojska-Dzienio

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 1, page 21-34
  • ISSN: 0010-2628

Abstract

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Medial modes, a natural generalization of normal bands, were investigated by Płonka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by Pöschel and Reichel. In this paper we show that each medial mode embeds as a subreduct into a semimodule over a certain ring, and that a similar theorem holds for each Lallement sum of cancellative modes over a medial mode. Similar results are obtained for rectangular algebras. The paper generalizes earlier results of A. Romanowska, J.D.H. Smith and A. Zamojska-Dzienio.

How to cite

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Zamojska-Dzienio, Anna. "Medial modes and rectangular algebras." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 21-34. <http://eudml.org/doc/249873>.

@article{Zamojska2006,
abstract = {Medial modes, a natural generalization of normal bands, were investigated by Płonka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by Pöschel and Reichel. In this paper we show that each medial mode embeds as a subreduct into a semimodule over a certain ring, and that a similar theorem holds for each Lallement sum of cancellative modes over a medial mode. Similar results are obtained for rectangular algebras. The paper generalizes earlier results of A. Romanowska, J.D.H. Smith and A. Zamojska-Dzienio.},
author = {Zamojska-Dzienio, Anna},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {modes (idempotent and entropic algebras); cancellative modes; sums of algebras; embeddings; semimodules over semirings; idempotent subreducts of semimodules; modes; cancellative modes; embeddings; semimodules over semirings},
language = {eng},
number = {1},
pages = {21-34},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Medial modes and rectangular algebras},
url = {http://eudml.org/doc/249873},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Zamojska-Dzienio, Anna
TI - Medial modes and rectangular algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 21
EP - 34
AB - Medial modes, a natural generalization of normal bands, were investigated by Płonka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by Pöschel and Reichel. In this paper we show that each medial mode embeds as a subreduct into a semimodule over a certain ring, and that a similar theorem holds for each Lallement sum of cancellative modes over a medial mode. Similar results are obtained for rectangular algebras. The paper generalizes earlier results of A. Romanowska, J.D.H. Smith and A. Zamojska-Dzienio.
LA - eng
KW - modes (idempotent and entropic algebras); cancellative modes; sums of algebras; embeddings; semimodules over semirings; idempotent subreducts of semimodules; modes; cancellative modes; embeddings; semimodules over semirings
UR - http://eudml.org/doc/249873
ER -

References

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  1. Ježek J., Kepka T., Medial Groupoids, Rozpravy ČSAV, Rada Mat. Přírod. Věd. 93 2 (1983), 93 pp. (1983) MR0734873
  2. Kearnes K., Semilattice modes I: the associated semiring, Algebra Universalis 34 (1995), 220-272. (1995) Zbl0848.08005MR1348951
  3. Kuras J., Application of Agassiz Systems to Representation of Sums of Equationally Defined Classes of Algebras (in Polish), Ph.D. Thesis, M. Kopernik University, Toruń, 1985. 
  4. Płonka J., Diagonal algebras, Fund. Math. 58 (1966), 309-321. (1966) MR0194378
  5. Płonka J., On a method of construction of abstract algebra, Fund. Math. 61 (1967), 183-189. (1967) MR0225701
  6. Płonka J., A representation theorem for idempotent medial algebras, Fund. Math. 61 (1967), 191-198. (1967) MR0225702
  7. Płonka J., Romanowska A.B., Semilattice sums, in Universal Algebra and Quasigroup Theory (eds. A.B. Romanowska and J.D.H. Smith), Heldermann, Berlin, 1992, pp.123-158. MR1191225
  8. Pöschel R., Reichel M., Projection algebras and rectangular algebras, in General Algebra and Applications (eds. K. Denecke and H.-J. Vogel), Heldermann, Berlin, 1993, pp.180-194. MR1209898
  9. Romanowska A.B., Smith J.D.H., Modal Theory, Heldermann, Berlin, 1985. Zbl0553.08001MR0788695
  10. Romanowska A.B., Smith J.D.H., Embedding sums of cancellative modes into functorial sums of affine spaces, in Unsolved Problems on Mathematics for the 21st Century, a Tribute to Kiyoshi Iseki's 80th Birthday (eds. J.M. Abe and S. Tanaka), IOS Press, Amsterdam, 2001, pp.127-139. Zbl0989.08001MR1896671
  11. Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002. Zbl1060.08009MR1932199
  12. Romanowska A.B., Semi-affine modes and modals, Sci. Math. Jpn. 61 (2005), 159-194. (2005) Zbl1067.08001MR2111551
  13. Romanowska A.B., Traina S., Algebraic quasi-orders and sums of algebras, Discuss. Math. Algebra Stochastic Methods 19 (1999), 239-263. (1999) Zbl0949.08001MR1709970
  14. Romanowska A.B., Zamojska-Dzienio A., Embedding semilattice sums of cancellative modes into semimodules, Contributions to General Algebra 13 (2001), 295-304. (2001) Zbl0993.08004MR1854593
  15. Romanowska A.B., Zamojska-Dzienio A., Embedding sums of cancellative modes into semimodules, Czechoslovak Math. J. 55 4 (2005), 975-991. (2005) Zbl1081.08003MR2184378
  16. Zamojska-Dzienio A., Embedding modes into semimodules (in English), Ph.D. Thesis, Warsaw University of Technology, Warszawa, 2003. 

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