We can define the notion of self-duality on lattices as follow :
A lattice $L$ is self-dual if there is a permutation $\pi$ such that : for all $a \in L$, $\ \downarrow \pi(a)$ is antiisomorphic to $\uparrow a$ .
Thus one might consider the "dual definition" of self-duality as follow :
A lattice $L$ is "anti-self-dual" if there is a permutation $\pi$ such that : for all $a \in L$, $\downarrow \pi(a)$ is isomorphic to $\uparrow a$
I am enable to find any literature about the second definition. Does it appear somewhere ?
The notion of self-duality can be extended to other structures like posets. Can the notion of "anti-self-duality" be extended on other structures ?
The notion of anti-self-duality seems harder to control than the notion of self-duality. It's much harder to find anti-self-dual lattices than self-dual one. Moreover I am aware of a self-dual lattice without any permutation $\pi$ (that respect the definition of self-duality) of order 2 whereas I am enable at the moment to find an anti-self-dual lattice without a permutation $\pi$ (that respect the definition of anti-self-duality) of order $2$. So why would you expect the notion of anti-self-dual to be much harder to control/understand ?
Thank you.