I'm not sure if this is of any interest at all, but I spent some time looking at it a couple of years ago so I'd like to ask for input on this.
Given two binary relations $\rho,\,\sigma$ on a set $X,$ we can compose them by the standard definition:
$$\rho\circ\sigma=\{(a,b)\in X\times X\,|\,(\exists\,c\in X)((a,c)\in\rho\wedge(c,b)\in\sigma)\}.$$
This can be seen as a composition of endomorphisms of the "semi vector space" (or vector semispace, or whatever it's called) $\Bbb B^n,$ where $n=|X|$ and $\Bbb B$ is the semifield $(\{0,1\},\vee,\wedge).$ This is seen by identifying subsets of $X$ with binary vectors by their characteristic function and considering the action of binary relations on $2^X$ by images. In particular, this shows that this composition distributes over the union operation.
Now we can De Morgan-flip the definition and take
$$\rho\star\sigma=\{(a,b)\in X\times X\,|\,(\forall\,c\in X)((a,c)\in\rho\vee(c,b)\in\sigma)\}.$$
I've never seen this operation defined anywhere. $2^{X\times X}$ with $\star$ is isomorphic to $2^{X\times X}$ with $\circ$ by the complement function. Similarly, it can be viewed as the space of endomorphisms of $\tilde{\Bbb B}^n,$ where $\tilde{\Bbb B}$ is the semifield $(\{0,1\},\wedge,\vee)$ (where the additive operation is written first). Consequently, $\star$ distributes over intersection.
Is there any law connecting $\circ$ and $\star$? Distributive laws do not work. I've been trying to see if $\rho\circ(\sigma\star\tau)\subseteq(\rho\circ\sigma)\star(\rho\circ\tau),$ but no. It does seem that it holds for almost all triples $\rho,\sigma,\tau$ (by computer calculations up to n=20) but there are exceptions.
Another question is if $\star$ has ever been defined and named.
