First, as I said, Grätzer’s quasifields are completely different from Dembowski’s.

It seems that strong pseudofields are almost the same as near-domains; the only difference may be that the order of addition is reversed. These things correspond in one-to-one fashion with sharply 2-transitive groups.

There also seems to be a connection between pseudofields and Grätzer quasifields; in both cases the correspondence to sharply 2-transitive groups is many-to-one. These objects may also be equivalent in some sense, once you have got around the fact that Tits uses addition while Grätzer uses subtraction. ]]>

So there are no examples of rank at most 4; we know just two of rank 5; and infinitely many of rank 6.

]]>Gordon: I have challenged James Mitchell to compute the endomorphism monoid of this graph. It might be interesting to see just what else is there. ]]>

Probably not of much significance, but your graph has other non-uniform endomorphisms too – for example, one with fibres of size 5,5,5,5,5,10,10 whose image is three triangles arranged in a path. (Presumably this is just a refinement of the one you’ve exhibited, but I haven’t checked that.)

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