Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Embedding general algebras into modules


Authors: Michał M. Stronkowski and David Stanovsky
Journal: Proc. Amer. Math. Soc. 138 (2010), 2687-2699
MSC (2010): Primary 08A05, 15A78, 16Y60
DOI: https://doi.org/10.1090/S0002-9939-10-10356-6
Published electronically: April 9, 2010
MathSciNet review: 2644885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of embedding general algebras into modules is revisited. We provide a new method of embedding, based on Ježek's embedding into semimodules. We obtain several interesting consequences: a simpler syntactic characterization of quasi-affine algebras, a proof that quasi-affine algebras without nullary operations are actually quasi-linear, and several facts regarding the ``abelian iff quasi-affine'' problem.


References [Enhancements On Off] (What's this?)

  • 1. Richard H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944), 19-52. MR 0009963 (5,229d)
  • 2. Ralph Freese and Ralph N. McKenzie, Residually small varieties with modular congruence lattices, Trans. Amer. Math. Soc. 264 (1981), no. 2, 419-430. MR 603772 (83d:08012a)
  • 3. -, Commutator theory for congruence modular varieties, London Mathematical Society Lecture Note Series, vol. 125, Cambridge University Press, Cambridge, 1987. MR 909290 (89c:08006)
  • 4. Jonathan S. Golan, Semirings and affine equations over them: Theory and applications, Mathematics and its Applications, vol. 556, Kluwer Academic Publishers Group, Dordrecht, 2003. MR 1997126 (2004j:16050)
  • 5. H. Peter Gumm, Algebras in permutable varieties: Geometrical properties of affine algebras, Algebra Universalis 9 (1979), no. 1, 8-34. MR 508666 (80d:08010)
  • 6. -, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. 45 (1983), no. 286. MR 714648 (85e:08012)
  • 7. Joachim Hagemann and Christian Herrmann, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel) 32 (1979), no. 3, 234-245. MR 541622 (80j:08006)
  • 8. Udo Hebisch and Hanns J. Weinert, Semirings: Algebraic theory and applications in computer science, Series in Algebra, vol. 5, World Scientific Publishing Co. Inc., River Edge, NJ, 1998, translated from the 1993 German original. MR 1704233 (2000g:16053)
  • 9. Christian Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 119-125. MR 534504 (80h:08011)
  • 10. David Hobby and Ralph N. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR 958685 (89m:08001)
  • 11. Jaroslav Ježek, Terms and semiterms, Comment. Math. Univ. Carolin. 20 (1979), no. 3, 447-460. MR 550447 (81e:08008)
  • 12. Keith A. Kearnes, A quasi-affine representation, Internat. J. Algebra Comput. 5 (1995), no. 6, 673-702. MR 1365197 (96k:08002)
  • 13. -, Idempotent simple algebras, Logic and algebra (Pontignano, 1994), Lecture Notes in Pure and Appl. Math., vol. 180, Dekker, New York, 1996, pp. 529-572. MR 1404955 (97k:08004)
  • 14. Keith A. Kearnes and Ágnes Szendrei, The relationship between two commutators, Internat. J. Algebra Comput. 8 (1998), no. 4, 497-531. MR 1663558 (2000e:08001)
  • 15. Miklós Maróti and Ralph N. McKenzie, Existence theorems for weakly symmetric operations, Algebra Universalis 59 (2008), no. 3-4, 463-489. MR 2470592
  • 16. Ralph N. McKenzie, Finite equational bases for congruence modular varieties, Algebra Universalis 24 (1987), no. 3, 224-250. MR 931614 (89j:08007)
  • 17. Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, lattices, varieties. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. MR 883644 (88e:08001)
  • 18. Ralph N. McKenzie and John Snow, Congruence modular varieties: Commutator theory and its uses, Structural theory of automata, semigroups, and universal algebra, NATO Sci. Ser. II Math. Phys. Chem., vol. 207, Springer, Dordrecht, 2005, pp. 273-329. MR 2210134 (2006k:08014)
  • 19. David C. Murdoch, Structure of abelian quasi-groups, Trans. Amer. Math. Soc. 49 (1941), 392-409. MR 0003427 (2,218b)
  • 20. Petr Němec and Tomáš Kepka, $ T$-quasigroups. I, II, Acta Univ. Carolin. Math. Phys. 12 (1971), no. 1, 39-49 (1972); ibid. 12 (1971), no. 2, 31-49 (1972). MR 0320206 (47:8745)
  • 21. Robert W. Quackenbush, Quasi-affine algebras, Algebra Universalis 20 (1985), no. 3, 318-327. MR 811692 (87d:08006)
  • 22. Anna B. Romanowska and Jonathan D. H. Smith, Embedding sums of cancellative modes into functorial sums of affine spaces, Unsolved problems on mathematics for the 21st century, IOS, Amsterdam, 2001, pp. 127-139. MR 1896671 (2003c:08001)
  • 23. -, Modes, World Scientific Publishing Co. Inc., River Edge, NJ, 2002. MR 1932199 (2003i:08001)
  • 24. Jonathan D. H. Smith, Mal'cev varieties, Lecture Notes in Mathematics, vol. 554, Springer-Verlag, Berlin, 1976. MR 0432511 (55:5499)
  • 25. -, An introduction to quasigroups and their representations, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2268350 (2008a:20104)
  • 26. Michał M. Stronkowski, On embeddings of entropic algebras, Ph.D. thesis, Warsaw University of Technology, Warsaw, 2006.
  • 27. -, Cancellation in entropic algebras, Algebra Universalis 60 (2009), no. 4, 439-468. MR 2504751
  • 28. -, Embedding entropic algebras into semimodules and modules, International Journal of Algebra and Computation 19 (2009), no. 8, 1025-1047.
  • 29. Ágnes Szendrei, Modules in general algebra, Contributions to general algebra, 10 (Klagenfurt, 1997), Heyn, Klagenfurt, 1998, pp. 41-53. MR 1648809 (99m:08019)
  • 30. Kôshichi Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221-227. MR 0014105 (7,241g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 08A05, 15A78, 16Y60

Retrieve articles in all journals with MSC (2010): 08A05, 15A78, 16Y60


Additional Information

Michał M. Stronkowski
Affiliation: Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Warsaw, Poland – and – Eduard Čech Center, Charles University, Prague, Czech Republic
Email: m.stronkowski@mini.pw.edu.pl

David Stanovsky
Affiliation: Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Email: stanovsk@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-10-10356-6
Keywords: Quasi-linear algebras, quasi-affine algebras, abelian algebras
Received by editor(s): August 14, 2009
Received by editor(s) in revised form: November 29, 2009
Published electronically: April 9, 2010
Additional Notes: The first author was supported by the Eduard Čech Center Grant LC505 and by the Statutory Grant of Warsaw University of Technology 504G11200112000
The second author was supported by the institutional grant MSM 0021620839 and by the GAČR Grant #201/08/P056.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society