Three terms that seem to cause confusion: Idempotence, Hysteresis and Deterministic. The subject came up during my webcast on the Five Kinds of Python Functions. We can use all three terms to describe a function. Two of them are relevant to common design questions in software. The third is a higher-order architectural consideration, and not really part of writing a function definition in Python.
Idempotent -- narrowly defined -- means that f(f(x)) = f(x). In computer science, the meaning is stretched so that we can distinguish functions like random number generators from other functions. A random number generator (Python's random.random(), for example) is not idempotent. It returns a different result each time it's called. Many other functions are idempotent because they always return the same result given the same arguments. The functions in Python's os module are not idempotent. The results change based on external events.
Hysteresis is memory of previous events. A random number generator may have some hidden hysteresis so that it can return the next value in it's random sequence. Note that os.random() is explicitly based on /dev/random; it involves hysteresis where entropy is accumulated. A function that has an internal memoization cache has hysteresis; it will compute subsequent results more quickly after having memorized previous results.
Most functions are simply idempotent and don't generally involve any hysteresis. The result value is a fixed mapping from the argument values.
A memoization cache preserves idempotence while adding hysteresis. The functools.lru_cache decorator, for example, adds hysteresis to an otherwise idempotent function. The result value still reflects a fixed mapping.
A random number generator cannot have idempotence; it will leverage hysteresis. We should think of a random number generator as an iterator through a sequence of numbers. Given a seed, the sequence is entirely fixed. Between any two numbers, it's very difficult to predict one from the other without also knowing the hidden seed value.
We use idempotence and hysteresis to describe programming language features. These features are entirely deterministic. There's no room for arbitrary, undefined, unpredictable behavior. Any non-determinism needs to be very tightly boxed in by otherwise deterministic programming constructs.
When writing a Python function, we assume that the language is deterministic. Without this assumption, it's hard to conceive of what the language would be like. What if statements were executed out of the order we wrote them?
External events -- interrupts and the like -- are non-deterministic. They can happen at any time, with no relationship to what's going on inside the software world. Inside the software world, however, we expect that everything is deterministic. This means that a server must always cope with non-deterministic request ordering. Once request processing starts, however, we generally rely on essential non-deterministic software to process the results perfectly consistently.
An important example of bounded non-determinism is in Dijksta's hypothetical programming language described in A Discipline of Programming. Here there is explicit non-determinism among the "elif" and "eldo" clauses. The selection among true alternatives was specifically non-deterministic. Indeed, an evil demon would always strive to select the worst possible choice. There was no "first one that's true" kind of silliness that would allow certain kind of logic errors to survive.
A multiprocessing application leverages the OS to keep all processes separate. Each process can then operate deterministically. Any non-determistic OS events are concealed from the process by the normal OS libraries that generally queue up events into buffers.
A multithreaded application, however, has to use some kind of explicit synchronization to handle the inherent non-determinism of thread scheduling. Thread libraries make no guarantees about the exact sequence of operations between threads; the execution is non-deterministic between threads.
For real fun, read about the non-deterministic memory write orders. The Data Race article from Intel is illuminating. Google "non-deterministic memory write order" for interesting reading on how processors have gotten -- perhaps -- too complex to be completely trustworthy.
This is different, also, from "arbitrary." A word that describes how the C language deals with constructs like a[i]= i++;. There are two unrelated state changes that happen in this statement. The order of those two things is best described as "arbitrary." It's deterministic. But it's not well defined by the language. Depending on the compiler and optimization settings, it will be entirely fixed. A function that uses these constructs could be idempotent. However, the outcome can vary from compiler to compiler and optimization setting to optimization setting. This kind of thing is devoutly to be avoided. It's presence is a less-than-ideal language design choice; writing software around a bad language feature is simply inexcusable.