Jordan Isomorphisms of the Finitary Incidence Ring of a Partially Ordered Category

Let $\mathcal{C}$ be a pocategory, $FI(\mathcal{C})$ the finitary incidence ring of $\mathcal{C}$ and $\phi$ a Jordan isomorphism of $FI(\mathcal{C})$ onto an associative ring $A$. We study the problem of decomposition of $\phi$ into the (near-)sum of a homomorphism and an anti-homomorphism.