The notion of an Exact Upper Bound is well-known, see page 1155 at http://dutiaw37.twi.tudelft.nl/~kp/onderwijs/set_theory/material/14-abraham_magidor-cardinal_arithmetic.pdf Question: What is an example of, P, X and a, such that "a=supX" but which is not an Exact Upper Bound. Thank you, Jan Pax