# Partial Ordering and Covering Relations

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 $\supset$, $\supseteq$,$\le$,$<$?

• "there is no transitivity" is a little vague because it's not clear whether you're asking about the binary relation "covers" or the binary relation ≺. I tried to handle both cases in my answer. – Mark S. Nov 12 '12 at 21:48
• @helopticor I am sorry. The question about transitivity was in regards to the covering relation. – Mack Nov 13 '12 at 16:43

"A partial ordered set (poset) is just a relation on a set, right?" A partially ordered set is a set with a partial order. A (lax) partial order is a relation satisfying Reflexivity, Antisymmetry, and Transitivity. "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?" A covering relation has antisymmetry trivially, but never has reflexivity and almost never has transtivity. You should think of a covering relation as arising from a given poset. "Or is it just saying that there isn't an element between $x$ and $y$? But wouldn't that mean there is no transitivity?" y covers x when x≺y and there's nothing in between. "≺" still has transitivity because of things like "if z covers y and y covers x then x≺z (even though z doesn't cover x)", but "covers" isn't transitive (unless you're in a boring case like "$S$ has only two elements").
"Also, is the symbol $≺$ just a generalization of the different symbols used, such as $\supset$, $\supseteq$,$\le$,$<$?" Yes, it's different so you don't confuse it with any particular one of those in any particular context.