On ordering of surjective cardinals

Abstract

Let $\mathrm{Card}$ denote the class of cardinals. For all cardinals $\mathfrak{a}$ and $\mathfrak{b}$, $\mathfrak{a}⩽\mathfrak{b}$ means that there is an injection from a set of cardinality $\mathfrak{a}$ into a set of cardinality $\mathfrak{b}$, and $\mathfrak{a}{⩽}^{\ast }\mathfrak{b}$ means that there is a partial surjection from a set of cardinality $\mathfrak{b}$ onto a set of cardinality $\mathfrak{a}$. A doubly ordered set is a triple $⟨P,\preccurlyeq ,{\preccurlyeq }^{\ast }⟩$ such that $\preccurlyeq$ is a partial ordering on $P$, ${\preccurlyeq }^{\ast }$ is a preordering on $P$, and $\preccurlyeq \subseteq {\preccurlyeq }^{\ast }$. In 1966, Jech proved that for every partially ordered set $⟨P,\preccurlyeq ⟩$, there exists a model of $\mathsf{ZF}$ in which $⟨P,\preccurlyeq ⟩$ can be embedded into $⟨\mathrm{Card},⩽⟩$. We generalize this result by showing that for every doubly ordered set $⟨P,\preccurlyeq ,{\preccurlyeq }^{\ast }⟩$, there exists a model of $\mathsf{ZF}$ in which $⟨P,\preccurlyeq ,{\preccurlyeq }^{\ast }⟩$ can be embedded into $⟨\mathrm{Card},⩽,{⩽}^{\ast }⟩$.

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