Possible and impossible conditionals for team logics

We study whether a logic based on team semantics can be enriched with a conditional satisfying minimal requirements, such as preservation of the closure property of the logic, Modus Ponens, and the Deduction Theorem. We show that such well-behaved conditionals exist for downward or upward closed logics, but do not typically exist for union closed, convex or intersection closed logics. We also briefly investigate conditionals satisfying weaker requirements.