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Abstract: In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank. [Copyright &y& Elsevier]

We investigate the coradical filtration of pointed coalgebras. First, we generalize a theorem of Taft and Wilson using techniques developed by Radford. We then look at the coradical filtration of duals of inseparable field extensions L* upon extension of the base field K, where K⊆L is a field extension. We reduce the problem to the case that the field extension is purely inseparable. We use this to prove that if E is a field containing the normal closure of L over K, then E⊗L*=(E⊗L*)1 if and only if L/K is separable or char(K)=&z.sfnc;L:Ls&z.sfnc;=2, where Ls is the separable closure of K in L. [Copyright &y& Elsevier]