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Journal of Convex Analysis 22 (2015), No. 1, 259--289
Copyright Heldermann Verlag 2015

Some Remarks on an Idempotent and Non-Associative Convex Structure

Walter Briec
CAEPEM, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France

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 R+n, B-convexity was not defined with respect to a given explicit algebraic structure. This is done here by proposing an extension of 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 Rn is shown and an explicit extension of B-convexity is proposed.

Keywords: Idempotence, semilattices, generalized convexity, B-convexity.

MSC: 06D50, 32F17

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