Yes, obviously. In the absence of a property-expropriating agency (whether you call it a state, a mob, a brigand, etc.), property rights are manifestly respected. But this is a semantic answer for a semantic question. It doesn't teach us anything about the world and how it works, but about the tools we use to describe it.
I think you mean to ask
>After the abolition of central power, will rule of law persist among the former constituents?
This is where the science is.
Anyway, if this is actually what you're asking, then it truly depends on the social development of the region in question. Somalia is a good case study. Civil war has effectively emasculated the competing states, turning the place into a power vacuum. Somalia, due to its geography, climate, history, gene pool, etc. has no rule of law in its tradition. In the absence of a tax-defense-law monopoly, its constituents have not practiced and will continue not to practice rule of law.
Take a look at, say, ancient Iceland. The vikings who settled it, far from the kings they answered to, already had rule of law and property rights in their tradition. They came from centuries of development under the selective pressures of their homeland: Scandinavia. That is why the Commonwealth of Iceland, until it was conquered by Norway, was effectively a private-law society. Even today, Nordic citizens have a strong enough legal tradition that all their countries have relatively free economies, in spite of their incredibly large welfare programs. If the migrants weren't a problem, the Scandinavian and Icelandic governments could collapse today, and the citizens would still very likely respect each other's legal rights.