Construction of finite ternary rings (Q2276964)

It is known that to each projective plane P we can associate a ternary ring A, i.e. a mapping \(\Phi\) : \(A\times A\times A\to A\) satisfying some axioms, and conversely to each ternary ring we can associate a projective plane [\textit{M. Hall jun.}, Combinatorial theory (1967; Zbl 0196.024)]. Let G(F) be a Galois projective plane, where F is a field. The set of elements of F with the operation \(<abc>=ab+c\) is the ternary ring corresponding to this plane. Let \(\phi\) and \(\psi\) be any functions from F into F and define a new ternary operation \(<abc>_{\phi,\psi}=ab+\psi (b)c+\phi (a)\). The problem is to find conditions on \(\phi\) and \(\psi\) under which \(<abc>_{\phi,\psi}\) is a ternary ring. The main result is Theorem 1: Suppose the field F is finite. The ternary operation \(<abc>_{\phi,\psi}\) is a ternary ring if and only if for any element \(a\in F\) we have \(\psi\) (a)\(\neq 0\) and for the set \(A_{\phi}\) of all elements of F of the form \((\phi (a_ 1)-\phi (a_ 2))/(a_ 1-a_ 2),\) where \(a_ 1\in F\), \(a_ 2\in F\), \(a_ 1\neq a_ 2\), and for the set \(B_{\psi}\) of all elements of \(F\cup \infty\) of the form \((\psi (b_ 1)b_ 2-\psi (b_ 2)b_ 1)/(\psi (b_ 2)-\psi (b_ 1)),\) where \(b_ 1,b_ 2\in F\), \(b_ 1\neq b_ 2\), we have the relation \(A_{\phi}\cap B_{\psi}=\emptyset\). A way to construct functions \(\phi\) and \(\psi\) satisfying the conditions of this theorem is given, but it is not known whether there are fields F satisfying one of the propositions of the paper.