There are many universal computation systems. Turing machines, tag systems, rewrite systems, cellular automata to name just a few. The universality of a system is proved via reduction from a known universal (Turing-complete) model.
I am wondering if it is possible to define abstract conditions, that make each of the systems universal. For example, in definitions of algebraic structures, there are conditions (like: Closure, Associativity, etc.) that can be examined in a structure to determine if it is e.g. a group. There is no reduction from other structure, that is already known to be group.
Is it possible to define in similar way the universality? Are there any related works on the topic?