Give the elements of the partially ordered set,which covering relation is the following $\{(a,c),(c,d),(b,e),(d,e) \} \subseteq \{a,b,c,d,e \}^2$

Question

$ \{(a,c),(c,d),(b,e),(d,e) \} \subseteq \{a,b,c,d,e \}^2$

I don't know exactly, how to write these elements. I know that a poset has to be reflexive,antisymmetric, and transitive. I also know that a covering relation of a poset is a binary relation which holds between comparable elements that are immediate neighbours.

It's the Cartesian product of ${a,b,c,d,e}$ with itself. That is, it's the set of all ordered pairs of elements from ${a,b,c,d,e}$. — Fabio Somenzi, Apr 12 '17 at 17:40
So it will be ${a,b,c,d,e }^2={(a,a),(a,b),(a,c),(a,d),(a,e),(b,a)...(e,e)}$? — Herrpeter, Apr 12 '17 at 17:43
Yes, it's the set of all 25 distinct pairs. Every binary relation on ${a,b,c,d,e}$ is a subset of ${a,b,c,d,e}^2$. — Fabio Somenzi, Apr 12 '17 at 17:46
I have to give the elements of the poset, which covering relation is ${(a,c),(c,d),(b,e),(d,e) } \subseteq {a,b,c,d,e }^2$ this is the text of the exercise — Herrpeter, Apr 12 '17 at 18:00

score 1 · Accepted Answer · answered Apr 12 '17 at 17:59

So ... I think your question is: $ \{(a,c),(c,d),(b,e),(d,e) \}$ is the covering relation of what poset?

To answer this, we need to make it reflexive, i.e. add $(a,a)$, $(b,b)$, etc., and transitive (i.e. find its transitive closure).

So, given that you have $(a,c)$, and $(c,d)$, you need to add $(a,d)$. Likewise, you need to add $(c,e)$, and therefore also $(a,e)$.

So you end up with:

$ \{(a,c),(c,d),(b,e),(d,e),(a,a),(b,b),(c,c),(d,d),(e,e),(a,d),(c,e),(a,e) \}$

Give the elements of the partially ordered set,which covering relation is the following $\{(a,c),(c,d),(b,e),(d,e) \} \subseteq \{a,b,c,d,e \}^2$

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