In recent work, we introduced a new semantics for conditionals, covering a large class of what we call preconditionals. In this paper, we undertake an axiomatic study of preconditionals and subclasses of preconditionals. We then prove that any bounded lattice equipped with a preconditional can be represented by a relational structure, suitably topologized, yielding a single relational semantics for conditional logics normally treated by different semantics, as well as generalizing beyond those semantics.