A Ring with Arithmetical Congruence Lattice Not Preserved by Any Pixley Function
Ivan Korec · Proceedings of the American Mathematical Society · 1981
A ring $(A; + , \cdot )$ is constructed such that the congruence lattice ${L_A}$ of the ring $(A; + , \cdot )$ is distributive, the elements of ${L_A}$ are pairwise permutable and there is no ${L_A}$-compatible function $p$ on $A$ such that \[ p(a,b,b) = p(a,b,a) = p(b,b,a) = a\quad {\text {for}}\;{\text {all}}\;a,b \in A.\] (1)