On Some Idempotent and Non-Associative Convex Structure

Abstract

$\mathbb{B}$-convexity was defined in [7] as a suitable Kuratowski-Painlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in the special case where convex sets are subsets of ${\mathbb{R}}_{+}^n$, $\mathbb{B}$-convexity was not defined with respect to a given explicit algebraic structure. This is done in that paper, which proposes an extension of $\mathbb{B}$-convexity to the whole Euclidean vector space. An unital idempotent and non-associative magma is defined over the real set and an extended $n$-ary operation is introduced. Along this line, the existence of the Kuratowski-Painlevé limit of the convex hull of two points over ${\mathbb{R}}^n$ is shown and an explicit extension of $\mathbb{B}$-convexity is proposed.