Abstract
LetX be a vector space over a totally ordered fieldF and Λ a subset ofF. Necessary and sufficient conditions are obtained on Λ for a join defined bya·b={λa+(1−λ)b|λ∈Λ} to make (X,·) (i) a convexity space (ii) a join system and (iii) a join space. For (X,·) to be a convexity space Λ is necessarily the set of elements between 0 and 1 for some ordered subfield ofF. In this setting join systems and join spaces are essentially the same but they remain quite distinct from convexity spaces.
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Baston, V.J., Bostock, F.A. Join operations in vector spaces. Monatshefte für Mathematik 83, 93–97 (1977). https://doi.org/10.1007/BF01534630
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DOI: https://doi.org/10.1007/BF01534630