Join operations in vector spaces

LetX be a vector space over a totally ordered fieldF and a subset ofF. Necessary and sufficient conditions are obtained on for a join defined bya·b={a+(1–)b|} to make (X,·) (i) a convexity space (ii) a join system and (iii) a join space. For (X,·) to be a convexity space 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.