I work in the algebra $R$ of reflexive, transitive relations over some set $S$, ordered by subset inclusion. This is a complete lattice, with intersection as g.l.b. and transitive hull as l.u.b., i.e. $r_1\vee r_2=( r_1\cup r_2)^+$, where ${}^+$ is transitive closure.

I'm interested in the residual $\ominus$ to $\vee$. Standard residual theory seems to tell me that $r_2\ominus r_1= \bigvee\{ r\mid r\vee r_1\subseteq r_2\}$.

Did I get this right? I'm not really on safe ground here... Also, this formulation is, well, somewhat indirect: does there exist a more direct formula?

Edit: To perhaps de-mystify this a bit, let me give the definition of residual: $\ominus$ is a residual to $\vee$ if it holds for all $r_1$, $r_2$, $r_3$ that $r_1\vee r_2\subseteq r_3$ iff $r_1\subseteq r_3\ominus r_2$. (Yes, this looks like adjoints in category theory; like adjoints, residuals are unique up to isomorphism if they exist.)


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