Cancellation laws for surjective cardinals (Q760421)

For cardinal numbers x and y, write \(x\leq^*y\) if whenever \(| X| =x\) and \(| Y| =y\) there is a map from a subset of Y onto X, and write \(x=^*y\) for \(x\leq^*y\) and \(y\leq^*x\). The \(=^*\)-equivalence classes are called surjective cardinals. This paper contains a number of interesting results concerning surjective cardinals in ZF set theory. (The axiom of choice is not assumed, of course.) Section 2 is devoted to showing that, with the ordering on \(=^*\)-equivalence classes defined by (x)\(\leq (y)\Leftrightarrow \exists z(x+z=^*y)\), the surjective cardinals form a weak cardinal algebra, as defined in an earlier paper of the author [Proc. London Math. Soc., III. Ser. 27, 577-599 (1973; Zbl 0272.02086)]. Section 3 establishes a number of cancellation laws, including the result \(kx=^*ky\Rightarrow x=^*y\) for any positive integer k. In Section 5 it is shown that the cancellation law \(kx\leq^*ky\Rightarrow x\leq^*y\) is unprovable in ZF. (The argument combines permutation methods and forcing.)