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Products of two sober directed complete posets need not be sober

Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao

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Source: Crossref

Published: Jun 1, 2026

DOI: 10.1112/jlms.70606

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Source abstract

Abstract We construct two directed complete posets 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. In addition, we show that if and are of a special type of sober complete lattices, then the Scott space of their order product is sober.

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