The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.
"Partition Principles and Infinite Sums of Cardinal Numbers." Notre Dame J. Formal Logic 36 (3) 425 - 434, Summer 1995. https://doi.org/10.1305/ndjfl/1040149358