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I am working on a article in poset theory. In that article, I am defining a subposet of a poset. The definition is following:

Let $P$ be a finite poset. A subposet $P'$ of $P$ is called closed under covering if for every $x,y \in P'$ with $x\lessdot y$ in $P'$, we have $x\lessdot y$ in $P$. Here, $x \lessdot y$ means $x$ is covered by $y$.

I want to know weather the above definition is already in the literature? If yes, then what these subposets are called? If not, then the name I have given is correct? or what should I call such subposets?

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    $\begingroup$ So, the Hasse diagram of the subposet is a subgraph of the Hasse diagram of the poset? $\endgroup$
    – Martin Rubey
    Commented Jul 1, 2021 at 5:47
  • $\begingroup$ Yes, the Hasse diagram of the subposet is a subgraph of the Hasse diagram of the poset. $\endgroup$
    – User007
    Commented Jul 1, 2021 at 6:02
  • $\begingroup$ Are you restricting to e.g. locally finite posets $P$ (for which the Hasse diagram is most relevant)? $\endgroup$
    – Sam Hopkins
    Commented Jul 1, 2021 at 6:06
  • $\begingroup$ Sorry, I forgot to mention. $P$ is a finite poset. I have edited the definition. $\endgroup$
    – User007
    Commented Jul 1, 2021 at 6:21

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I recall seeing in various sources the terminology "cover preserving embedding" and "cover preserving subposet". Googling it now (https://www.google.com/search?q=poset+%22cover+preserving%22) brings some 4000 results, many of which are research articles (with some repetitions - I am not implying there are 4000 distinct articles on this topic).

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    $\begingroup$ Thank you, the terminology "cover preserving subposet" is being used in many articles. $\endgroup$
    – User007
    Commented Jul 1, 2021 at 6:44

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