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Biordered sets and complemented modular lattices

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Abstract

In this paper we give a structure theorem for the biordered set of a strongly regular Baer semigroup. As a result we shall be able to construct the biordered set of the multiplicative semigroup of a regular ring in terms of the complemented modular lattice which is coordinatized by this regular ring.

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References

  1. Birkhoff, G.,Lattice Theory, Providence (1967).

  2. Blyth, T.S. and M.F. Janowitz,Residuation Theory, Oxford (1972).

  3. Chaptal, N.,Demi-groupe multiplicatif d'un anneau, C.R. Acad. Sci. Paris 262 (1966) 274–277.

    MATH  MathSciNet  Google Scholar 

  4. Clifford, A.H.,The partial groupoid of idempotents of a regular semigroup, Semigroup Forum 20 (1975) 262–268.

    Article  MathSciNet  Google Scholar 

  5. Clifford, A.H.,The fundamental representation of a regular semigroup, Semigroup Forum 10 (1975) 84–92.

    Article  MATH  MathSciNet  Google Scholar 

  6. Clifford, A.H. and G.B. Preston,The Algebraic Theory of Semigroups, Providence (1961).

  7. Dubreil-Jacotin, M.L., L. Lesieur et R. Croisot,Leçons sur la Théorie des Treillis,des Structures Algébriques Ordonnées et des Treillis Géométriques, Paris (1953).

  8. Dyer-Bennet, J.,A note on finite regular rings, Bull. Amer. Math. Soc., 47 (1941), 784–787.

    Article  MATH  MathSciNet  Google Scholar 

  9. Erdos, J.A.,On products of idempotent matrices, Glasgow Math. J. 8 (1967) 118–122.

    MATH  MathSciNet  Google Scholar 

  10. Fitz-Gerald, D.G.,On inverses of products of idempotents in regular semigroups, J. Australian Math. Soc. 13 (1972) 335–337.

    Article  MATH  MathSciNet  Google Scholar 

  11. Grillet, P.A.,The Structure of regular semigroups, I:a representation, Semigroup Forum 8 (1974) 177–183.

    Article  MATH  MathSciNet  Google Scholar 

  12. Grillet, P.A.,The structure of regular semigroups, II:cross-connections, Semigroup Forum 8 (1974) 254–259.

    Article  MATH  MathSciNet  Google Scholar 

  13. Janowitz, M.F.,Baer semigroups, Duke Math J. 32 (1965) 85–95.

    Article  MATH  MathSciNet  Google Scholar 

  14. Janowitz, M.F.,A semigroup approach to lattices, Canad. J. Math. 18 (1966) 1212–1223.

    MATH  MathSciNet  Google Scholar 

  15. Jonsson, B.,Representation of complemented modular lattices, Trans. Amer. Math. Soc. 97 (1960) 64–94.

    Article  MATH  MathSciNet  Google Scholar 

  16. Nambooripad, K.S.S.,Structure of Regular Semigroups, Dissertation, University of Kerala (1973).

  17. Nambooripad, K.S.S.,Structure of regular semigroups I:Fundamental regular semigroups, Semigroup Forum 9 (1975) 354–363.

    Article  MathSciNet  Google Scholar 

  18. Schein, B.M., O-rings and LA-rings, Izv. Vys. Uč. Zav. Mat. 2 (51) (1966) 111–122 (in Russian).

    Google Scholar 

  19. Schwarz, Š.,On dual semigroups, Czechoslovak Math. J. 10 (1960) 201–230.

    MathSciNet  Google Scholar 

  20. Schwarz, Š.,Any O-simple dual semigroup is completely O-simple, Semigroup Forum 2 (1971) 90–92.

    Article  MATH  MathSciNet  Google Scholar 

  21. Skornyakov, L.A.,Complemented Modular Lattices and Regular Rings, London (1964).

  22. von Neumann, J.,On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936) 707–713.

    Article  MATH  Google Scholar 

  23. von Neumann, J.,Algebraic theory of continuous geometries, Proc. Nat. Acad. Sci. Wash. 23 (1937) 16–22.

    Article  MATH  Google Scholar 

  24. von Neumann, J.,Continuous Geometry, Princeton (1960).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Nebraska-Lincoln, 68588, Lincoln, Nebraska, U.S.A.

    Francis Pastijn

  2. Dienst Hogere Meetkunde, Rijksuniversiteit te Gent, Krijgslaan 271, B-9000, Gent, Belgium

    Francis Pastijn

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  1. Francis Pastijn
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Additional information

Communicated by R. McFadden

The author's research was done while he was a visiting professor at the University of Nebraska, supported by a Fulbright-Hays Award.

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Pastijn, F. Biordered sets and complemented modular lattices. Semigroup Forum 21, 205–220 (1980). https://doi.org/10.1007/BF02572551

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  • DOI: https://doi.org/10.1007/BF02572551

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