Another Problem of J\'onsson and McKenzie from 1982: Refinement Properties for Connected Powers of Posets

Abstract

In 1982, J\'onsson and McKenzie posed the following problem: "Find counter examples (or prove that none exist) to the refinement of $A^C\cong B^D$ [$A$, $B$, $C$, and $D$ non-empty posets] under" the condition "$C$, $D$, and $A^C$ are finite and connected." That is, in this situation, are there posets $E$, $X$, $Y$, and $Z$ such that $A\cong E^X$, $B\cong E^Y$, $C\cong Y\times Z$, and $D\cong X\times Z$? In this note, this problem is solved.