On Some Idempotent and Non-Associative Convex Structure

Walter Briec

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arXiv · 1311.0690

On Some Idempotent and Non-Associative Convex Structure

Walter Briec

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Abstract

$\mathbb B$-convexity was defined in [7] as a suitable Kuratowski-Painlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in 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 in that paper, which proposes 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.

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Walter Briec. 2013-11-04. On Some Idempotent and Non-Associative Convex Structure. https://arxiv.org/abs/1311.0690

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