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

TL;DR

This paper addresses a longstanding open problem from 1982 by providing a solution to the question of refinement properties for connected powers of posets, specifically regarding isomorphisms of exponential posets.

Contribution

The paper solves the open problem posed by Jönsson and McKenzie in 1982 concerning refinement properties of connected powers of posets.

Findings

Established conditions under which the refinement property holds for connected powers of posets.

Provided counterexamples or proofs of non-existence for certain poset isomorphisms.

Clarified the structure of posets related to exponential isomorphisms in finite connected cases.

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.