When are the complemented elements of a bounded lattice neutral? Given any bounded lattice $\mathcal{L}=(X,\lor,\land,0,1)$ we say any $a\in X$ is complemented if there exists an element $b\in X$ such that $a\lor b=1$ and $a\land b=0$ likewise we refer to any element $a\in X$ as neutral when we have $\forall x,y\in X[(a\land x)\lor (a\land y)\lor (x\land y)$$=(a\lor x)\land (a\lor y)\land (x\lor y)]$
Now with that said, for which bounded lattices must all complemented elements be neutral?
Do these lattices have a name? Can they be characterized easily in terms of other lattices?
 A: I don't have a good answer to this, but let me record some observations.
First, Birkhoff's 1930's definition of a neutral element of a lattice is: $a\in L$ is neutral if, for any $x, y\in L$, the sublattice of $L$ generated by $\{a,x,y\}$ is distributive. It is Gratzer's 1960's theorem that this property is equivalent to $\forall x \forall y[(a\land x)\lor (a\land y)\lor (x\land y)$$=(a\lor x)\land (a\lor y)\land (x\lor y)]$.
An element $a\in L$ is neutral and complemented iff the map $L\to [a,1]\times [0,a]: x\mapsto (x\vee a, x\wedge a)$ is a lattice isomorphism. That is, neutral complements `determine' direct factorizations of $L$.
This is analogous to something that happens in ring theory, where one is concerned with idempotents ($e\in R$ satisfying $e^2=e$). Idempotent elements of a ring are complemented in the sense that $e\in R$ is idempotent iff there is an element $f$ such that $e+f=1$ and $e\cdot f = 0$.
Those idempotents that correspond to direct factorizations, $R\cong R/(e)\times R/(1-e)$, are precisely the central idempotents ($\forall r(re=er)$.)
So the analogy (perhaps it is something even deeper than analogy) goes like this:
idempotent of a ring corresponds to complemented element of a lattice, and central idempotent corresponds to neutral complement. If you wanted to continue this analogy, you might ask: What is the name for a ring whose idempotents are all central? (Maybe the same name could be used to refer to lattices whose complemented elements are neutral?)
Unfortunately, a ring whose idempotents are all central is called an abelian ring. This is a terrible, terrible choice of terminology, which we should not propagate by defining an abelian lattice to be a lattice whose complemented elements are all neutral.
