TITLE: A definition in poset theory
QUESTION [10 upvotes]: 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?

REPLY [10 votes]: 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).