A note on surjective cardinals

For cardinals a and b, we write a=^ if there are sets A and B of cardinalities a and b, respectively, such that there are partial surjections from A onto B and from B onto A. =^-equivalence classes are called surjective cardinals. In this article, we show that ZF+DC_, where is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss J. Truss, Ann. Pure Appl. Logic 27, 165--207 (1984). Nevertheless, we show that surjective cardinals form a ``surjective cardinal algebra'', whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that m=^ m implies a=^ for all cardinals a, b and all nonzero natural numbers m.