Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Walter Briec

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

Keywords

  • Idempotence
  • semilattices
  • generalized convexity
  • B-convexity

Mathematics Subject Classification

06D50, 32F17

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Suggested citation

W. Briec. “Some Remarks on an Idempotent and Non-Associative Convex Structure.” Journal of Convex Analysis 22 (2015), No. 1, 259–289. https://doi.org/10.68381/jca22013

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.