Some Remarks on an Idempotent and Non-Associative Convex Structure

$\mathbb{B}$-convexity was recently defined by the author and C. D. Horvath [B-convexity, Optimization 53(2) (2004) 103-127] as a suitable Kuratowski-Painlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Except for 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 here by proposing 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.