Separation Properties in some Idempotent and Symmetrical Convex Structure

B-convexity was 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. Recently, an alternative formulation over the whole Euclidean vector space was proposed [W. Briec, Some remarks on an idempotent and non-associative convex structure, Journal of Convex Analysis 22 (2015) 259–289]. In this paper a convex separation framework is proposed as well as some extension of known results established over posets. We first analyze the algebraic properties of some class of subsets characterized by a suitable notion of dual form. Along this line some extended separation results are established by considering the Kuratowski-Painlevé limit of a sequence of linear halfspaces.