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?



A definition in poset theory
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).


