Generators of existence varieties of regular rings and complemented Arguesian lattices

Christian Herrmann; Marina Semenova

Open Mathematics (2010)

  • Volume: 8, Issue: 5, page 827-839
  • ISSN: 2391-5455

Abstract

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We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.

How to cite

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Christian Herrmann, and Marina Semenova. "Generators of existence varieties of regular rings and complemented Arguesian lattices." Open Mathematics 8.5 (2010): 827-839. <http://eudml.org/doc/268988>.

@article{ChristianHerrmann2010,
abstract = {We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.},
author = {Christian Herrmann, Marina Semenova},
journal = {Open Mathematics},
keywords = {Complemented modular lattice; Regular ring; Existence variety; Matrix ring; complemented modular lattice; von Neumann regular ring; existence variety; matrix ring; complemented Arguesian lattice; continuous geometry; coordinatization of lattice geometries},
language = {eng},
number = {5},
pages = {827-839},
title = {Generators of existence varieties of regular rings and complemented Arguesian lattices},
url = {http://eudml.org/doc/268988},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Christian Herrmann
AU - Marina Semenova
TI - Generators of existence varieties of regular rings and complemented Arguesian lattices
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 827
EP - 839
AB - We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.
LA - eng
KW - Complemented modular lattice; Regular ring; Existence variety; Matrix ring; complemented modular lattice; von Neumann regular ring; existence variety; matrix ring; complemented Arguesian lattice; continuous geometry; coordinatization of lattice geometries
UR - http://eudml.org/doc/268988
ER -

References

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  7. [7] Herrmann C., Generators for complemented modular lattices and the von Neumann-Jónsson coordinatization theorems, Algebra Universalis, (in press), DOI: 10.1007/s00012-010-0064-5 Zbl1205.06004
  8. [8] Herrmann C., Huhn A.P., Zum Begriff der Charakteristik modularer Verbände, Math. Z., 1975, 144(3), 185–194 http://dx.doi.org/10.1007/BF01214134 Zbl0316.06006
  9. [9] Herrmann C., Semenova M., Existence varieties of regular rings and complemented modular lattices, J. Algebra, 2007, 314(1), 235–251 http://dx.doi.org/10.1016/j.jalgebra.2007.01.038 Zbl1139.06003
  10. [10] Jipsen P., Rose H., Varieties of Lattices, Lecture Notes in Mathematics, 1533, Springer, Berlin, 1992 Zbl0779.06005
  11. [11] Jónsson B., Representations of complemented modular lattices, Trans. Amer. Math. Soc., 1960, 97, 64–94 http://dx.doi.org/10.2307/1993364 Zbl0101.02204
  12. [12] Kad’ourek J., Szendrei M.B., On existence varieties of E-solid semigroups, Semigroup Forum, 1999, 59(3), 470–521 http://dx.doi.org/10.1007/s002339900066 
  13. [13] Maeda F., Kontinuierliche Geometrien, Die Grundlehren der mathematischen Wissenschaften, 95, Springer, Berlin, 1958 
  14. [14] Mal’cev A.I., Algebraic Systems, Die Grundlehren der mathematischen Wissenschaften, 192, Springer, Berlin, 1973 
  15. [15] von Neumann J., Continuous Geometry, Princeton Mathematical Series, 25, Princeton University Press, Princeton, 1960 
  16. [16] Skornyakov L.A., Complemented Modular Lattices and Regular Rings, Oliver & Boyd, Edinburgh-London, 1964 Zbl0156.04101
  17. [17] Veblen O., Young J.W., Projective Geometry I, Ginn & Co., Boston, 1910 

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