Reflexive Transitive Closure The problem I am working on is, "Show that a finite poset can be reconstructed from its covering relation. [Hint:Show that the poset is the reflexive transitive closure of its covering relation.]" I have been searching through my textbook, and on the internet, for the definition of reflexive transitive closure, but I was not successful. Could someone please explain this concept to me?

Suppose that $R$ is a relation on a set $A$. The reflexive transitive closure of $R$ is the smallest relation $S$ on $A$ such that

1. $R\subseteq S$;
2. $S$ is reflexive; and
3. $S$ is transitive.



If $R$ is already reflexive and transitive, then $R$ is its own reflexive transitive closure, but that’s not the case with your covering relations.

One way of constructing the reflexive transitive closure of $R$ is to begin by expanding $R$ to $$R_r=R\cup\\{\langle a,a\rangle:a\in A\\}\;,$$ adding to $R$ all of the pairs $\langle a,a\rangle$ that aren’t already in it. Then whenever you have $\langle x,y\rangle$ and $\langle y,z\rangle$ in $R_r$, you throw in $\langle x,z\rangle$ if it’s not already there to get $R_r^2$. Repeat to get $R_r^3$. If $A$ is finite, after a finite number of steps nothing new will be added; the resulting relation $R_r^*$ is the reflexive transitive closure of $R$.