Products of two sober dcpo's need not be sober

TL;DR

This paper demonstrates that the product of two sober dcpos may not be sober, providing a counterexample to an open problem, while also identifying conditions under which the product remains sober.

Contribution

It constructs specific counterexamples showing non-sobriety of product dcpos and establishes conditions for sobriety in products of certain sober lattices.

Findings

Counterexamples of non-sober product dcpos

Conditions ensuring sobriety in products of sober lattices

Resolution of an open problem on Scott space sobriety

Abstract

We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober.