Algebra universalis

, 79:54 | Cite as

Representability of Lyndon–Maddux relation algebras

  • Jeremy F. Alm
Part of the following topical collections:
  1. In memory of Bjarni Jónsson


In 2016, Alm–Hirsch–Maddux defined relation algebras \(\mathfrak {L}(q,n)\) that generalize Roger Lyndon’s relation algebras from projective lines, so that \(\mathfrak {L}(q,0)\) is a Lyndon algebra. In that paper, it was shown that if \(q>2304n^2+1\), then \(\mathfrak {L}(q,n)\) is representable, and if \(q<2n\), then \(\mathfrak {L}(q,n)\) is not representable. In the present paper, we reduced this gap by proving that if \(q\ge n(\log n)^{1+\varepsilon }\), then \(\mathfrak {L}(q,n)\) is representable.


Representable relation algebra Lovász local lemma 

Mathematics Subject Classification

03G15 05D40 


  1. 1.
    Alm, J.F., Hirsch, R., Maddux, R.D.: There is no finite-variable equational axiomatization of representable relation algebras over weakly representable relation algebras. Rev. Symb. Log. 9(3), 511–521 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.H.: The Probabilistic Method, fourth edition. Wiley Series in Discrete Mathematics and Optimization. Wiley , Hoboken, NJ (2016)Google Scholar
  3. 3.
    Andréka, H., Maddux, R.D., Németi, I.: Splitting in relation algebras. Proc. Am. Math. Soc. 111(4), 1085–1093 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jónsson, B.: The theory of binary relations. In: Algebraic logic (Budapest, 1988), Colloq. Math. Soc. János Bolyai, vol. 54, pp. 245–292. North-Holland, Amsterdam (1991)Google Scholar
  5. 5.
    Jukna, S.: Extremal Combinatorics, 2nd edn. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Lyndon, R.C.: Relation algebras and projective geometries. Michigan Math. J. 8, 21–28 (1961)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsLamar UniversityBeaumontUSA

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