# Does this property of a function $f : 2^A \rightarrow A$ have a name?

I've got this property: For all $$S\in 2^A$$ and all partitions $$P$$ of $$S$$, $$f(S) = f\left(\left\{f(M)\mid M \in P\right\}\right)$$, i.e. $$f$$ maps a set of values to a single value and gives the same result whether we map the set "at once" or whether we map subsets first and then map the set of the results. Examples for $$f$$ would be sums of finite sets of numbers, or the least upper bound of subsets of a complete lattice.

So, if $$A = \mathbb{N}\cup\left\{\infty\right\}$$ and $$f(S) = \sum_{s \in S} s$$, the property is fulfilled, as, for example $$f(\{1,2,3\}) = f(\{f(\{1,2\}),f(\{3\})\}) = 3+3 = 6$$

Is that a well-known property? If so, what's it called so I can read up more on it?

• The lack of explicit quantifiers makes your property hard to understand. Does your property say the following? For every $S \in 2^A$ and for every partition $P$ of $S$, $f(S) = f\left(\left\{f(M)\mid M \in P\right\}\right)$ Commented Jul 4, 2023 at 15:46
• Yes, exactly. I'll amend the question. Commented Jul 4, 2023 at 16:22
• Use \sum if you are talking about a sum. Use \Sigma if you are writing in Greek. Commented Jul 4, 2023 at 16:30