I am currently reading about partial ordering and covering relations. I just want to be certain that I am understanding these concepts correctly. A partial ordered set (poset) is just a relation on a set, right? It's used to order the elements the relation is a set on? And for the covering relation, the way the author describes seems to indicate that a covering relation is very similar to the poset, except it doesn't have transitivity?
The book says, "We say that an element $y∈S$ covers an element $x∈S$ if $x≺y$ and there is no element $z∈S$ such that $x≺z≺y$." Or is it just saying that there isn't an element between $x$ and $y$? But wouldn't that mean there is no transitivity?
Also, is the symbol $≺$ just a generalization of the different symbols used, such as $\supset$, $\supseteq$,$\le$,$<$?