Keimel's problem on the algebraic axiomatization of convexity
In this paper, the authors study the following: real convex sets may be presented algebraically as sets with binary operations given by weighted means, the weights being taken from the open unit interval \(I\circ=]0,1[\) in the set of reals. The class of convex sets is a quasivariety, and generates the variety, defined by identities of so called barycentric algebras (see Section 2). Both these classes have a well-developed theory. In particular, like all modes, barycentric algebras implement self-distributivity in the sense of \textit{O. Frink} [Am. Math. Mon. 62, 697--707 (1955; Zbl 0065.26501)]. The variety of barycentric algebras contains convex sets, semilattices, and certain semilattice sums of convex sets. It is defined by three types of axioms: idempotence, skew commutativity, and skew-associativity, the latter two implying the entropic property that each operation is a homomorphism. Since the skew-associativity axiom is rather complicated, Klaus Keimel asked whether it may simply be replaced by entropicity alone. In this paper, the authors show that the answer is negative. They construct an entropic algebra, of the same type as barycentric algebras, which is idempotent and skew-commutative, but is not skew-associative (see Theorem 5.1). Together with barycentric algebras, the counterexample presented by authors belongs to a much larger family of algebras described as threshold barycentric algebras, which model the concept of threshold convexity introduced in Section 4. Thus threshold barycentric algebras offer an entire spectrum of algebras, ranging from the usual barycentric algebras at one end to the commutative binary modes at the other.
scientific article; zbMATH DE number 6904407
scientific article