Algebra universalis

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Keimel’s problem on the algebraic axiomatization of convexity

  • A. Komorowski
  • A. B. Romanowska
  • J. D. H. Smith
Open Access


Convex sets may be viewed as algebras equipped with a set of binary convex combinations that is indexed by the open unit interval of real numbers. Convex sets generate the variety of barycentric algebras, which also includes semilattices where the semilattice multiplication is repeated uncountably many times. Barycentric algebras are defined by three axioms: idempotence, skew-commutativity, and skew-associativity. Since the skew-associativity axiom is rather complicated, Klaus Keimel has asked whether it can simply be replaced by the entropic law. It turns out that the answer is negative. The counterexamples presented and studied in this paper are known as threshold barycentric algebras, depending on a threshold taken from the left-hand side of the closed unit interval. They offer an entire spectrum of algebras, ranging from barycentric algebras for threshold 0 to commutative idempotent entropic groupoids for threshold 1/2.


Entropic algebra Barycentric algebra Convex set Semilattice Self-distributive Convexity 

Mathematics Subject Classification

08A99 52A01 


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • A. Komorowski
    • 1
  • A. B. Romanowska
    • 1
  • J. D. H. Smith
    • 2
  1. 1.Faculty of Mathematics and Information SciencesWarsaw University of TechnologyWarsawPoland
  2. 2.Department of MathematicsIowa State UniversityAmesUSA

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