MathDB

Let

\mathcal{X}

be the collection of all non-empty subsets (not necessarily finite) of the positive integer set

\mathbb{N}

. Determine all functions

f: \mathcal{X} \to \mathbb{R}^+

satisfying the following properties:
(i) For all

S

T \in \mathcal{X}

with

S\subseteq T

, there holds

f(T) \le f(S)

. (ii) For all

S

T \in \mathcal{X}

, there hold f(S) + f(T) \le f(S + T), f(S)f(T) = f(S\cdot T), where

S + T = \{s + t\mid s\in S, t\in T\}

and

S \cdot T = \{s\cdot t\mid s\in S, t\in T\}

.
Proposed by Li4, Untro368, and Ming Hsiao.

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