# Is potential causality relation equals to set of consistent happens before relations?

In a book dedicated to distributed algorithms, I found the following definition - potential causality diagram is consistent with happens before diagram if potential causality relation between system events is a subset of happens-before relation. So far so good.

But then I have read following claim potential causality diagram equals to the set of happens before diagram to which it consistent with. But how that is possible? I mean that cardinality of potential causality relation is less than the cardinality of happens before relation to which it consistent with. How may potential causality relation be composed of union of such happens-before relations? I agree that potential causality relation may be composed of a set of happens-before relations, but from my point of view happens before relations in such case should be subsets of potential causality relations? Could you explain what I missed?

Update: Quotation from the book. "Given a potential causality diagram $(E, \xrightarrow{p})$, a happened before diagram $(E, \rightarrow)$ is consistent with it if $\xrightarrow{p}\subset \rightarrow$.

is consistent with Observe that $e_2 ->f_4$ but $\neg(e_2 \xrightarrow{p} f_4)$. A potential causality diagram is equivalent to the set of happened before diagrams that are consistent with it.

Vijay K. Garg. Elements of Distributed Computing (Kindle Locations 301-302). Kindle Edition.

• Welcome to CS.SE! Would it be possible to provide a citation from where you read that, and quote the relevant excerpt? It's possible that providing additional context might help someone give a better answer. – D.W. Jan 23 '17 at 14:08
• @D.W. I updated question. Thank you for suggestion ! – Andrey Lomakin Jan 23 '17 at 14:37