I'm generalizing/analogizing from the stuff I read on coalgebras, and in this case I'm pretty sure the idea makes sense, it's probably explored elsewhere. You can start here, or directly from Introduction to Coalgebra: Towards Mathematics of States and Observations (PDF).

There are many similarities (or dualities) between algebras and coalgebras which are often useful as guiding principles. But one should keep in mind that there are also significant differences between algebra and coalgebra. For example, in a computer science setting, algebra is mainly of interest for dealing with finite data elements – such as finite lists or trees – using induction as main definition and proof principle. A key feature of coalgebra is that it deals with potentially infinite data elements, and with appropriate state-based notions and techniques for handling these objects. Thus, algebra is about construction, whereas coalgebra is about deconstruction – understood as observation and modification.

A rule of thumb is: data types are algebras, and state-based systems are coalgebras. But this does not always give a clear-cut distinction. For instance, is a stack a data type or does it have a state? In many cases however, this rule of thumb works: natural numbers are algebras (as we are about to see), and machines are coalgebras. Indeed, the latter have a state that can be observed and modified.

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Initial algebras (in Sets) can be built as so-called term models: they contain everything that can be built from the operations themselves, and nothing more. Similarly, we saw that final coalgebras consist of observations only.