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I say that a relation $R$ on a poset $P$ is downward closed if for each $(x,y)\in R$, and $x'\le x$, then $(x',y)\in R$. Is there a reference where this thing is studied, maybe under a different name?

The motivation for this question is: I'm trying to prove or disprove that the lower/upper consistency relations, as defined in §1.3 here, have this property.

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    $\begingroup$ Downward closed sets are sometimes called "order ideals". $\endgroup$
    – Goldstern
    Commented Apr 23, 2016 at 17:42
  • $\begingroup$ This is just a set-mapping $P\to D(P)$, where $D(P)$ is the set of all downward closed sets of $P$. Never heard of anyone studying this strange kind of animal. However, if you would want to know something about isotone mappings $P\to D(P)$, that would be another story, of course. $\endgroup$
    – Gejza Jenča
    Commented Apr 25, 2016 at 17:44
  • $\begingroup$ Relations for which $x'\leq x R y\leq y'\Longrightarrow x' R y'$ are most certainly a very common kind of animal. Mappings $X\to T X$ for any endofunctor of any category are another very common kind of animal, called a coalgebra, which enough people study to run a conference series. $\endgroup$
    – Paul Taylor
    Commented Apr 26, 2016 at 21:44

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