
Journal of Convex Analysis 22 (2015), No. 1, 259289 Copyright Heldermann Verlag 2015 Some Remarks on an Idempotent and NonAssociative Convex Structure Walter Briec CAEPEM, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France Bconvexity was recently defined by the author and C. D. Horvath [Bconvexity, Optimization 53(2) (2004) 103127] as a suitable KuratowskiPainlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Except for the special case where convex sets are subsets of R_{+}^{n}, Bconvexity was not defined with respect to a given explicit algebraic structure. This is done here by proposing an extension of Bconvexity to the whole Euclidean vector space. An unital idempotent and nonassociative magma is defined over the real set and an extended nary operation is introduced. Along this line, the existence of the KuratowskiPainlevé limit of the convex hull of two points over R^{n} is shown and an explicit extension of Bconvexity is proposed. Keywords: Idempotence, semilattices, generalized convexity, Bconvexity. MSC: 06D50, 32F17 [ Fulltextpdf (300 KB)] for subscribers only. 