Have you seen this sort of an anti-involution on a lattice?

While looking at a representation theory question, I came up with the following sort of object. I want to know if it comes up often in combinatorics or some other area of mathematics.

Let $P$ be a finite poset and $a:P\to P$ be an order-reversing bijection. Then the lattice $J(P)$ of order ideals in $P$ gets an order reversing bijection by $I\mapsto (P-I^a)$, where $I^a$ denotes the image of $I$ under ${a}$.

For example, if $P$ is an antichain and $a$ is the identity, then this is a complement operation on the lattice. But in contrast, if $P$ is a chain, then you do not get a complemented lattice.

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You can for example check Combinatorics: The Rota Way, pp. 19-20 (Birkhoff equivalence between finite distributive lattices and finite posets) and p. 21. (duality between join and meet irreducibles). Conclusion: you are considering generic anti-automorphisms of finite distributive lattices. –  user46855 Feb 16 at 22:24
@user46855 So every lattice anti-automorphism of a finite distributive lattice is of this form. That's nice to know. Thank you. –  Amritanshu Prasad Feb 17 at 4:01