# Does every orthocomplemented lattice satisfy the shuffle laws?

Let $L=(X,\wedge,\vee,^\bot)$ denote a non-empty lattice having a unary operation $x \mapsto x^\bot$ that satisfies both "shuffle" laws: $$x \wedge y \leq z \iff x \leq y^\bot \vee z.$$

$$x \leq y \vee z \iff x \wedge y^\bot \leq z.$$

Then $L$ is an orthocomplemented lattice. (Proof below).

Question. Does every orthocomplemented lattice satisfy the shuffle laws? It suffices to prove just one of them, but I've been unable to do so.

Proof of claim.

Complements. Let $x$ be fixed but arbitrary. This is allowed, since $L$ was assumed non-empty. We will show that $x^\bot \wedge x$ is a least element. TFAE.

1. $x^\bot \wedge x \leq a$
2. $x^\bot \leq x^\bot \vee a$
3. TRUE

Thus $x^\bot \wedge x$ is a least element. A similar argument shows that $x^\bot \vee x$ is a maximum element.

Involutiveness. We will show that $x \leq x^{\bot\bot}.$ TFAE.

1. $x \leq a$
2. $x \wedge x \leq a$
3. $x \leq x^\bot \vee a$
4. $x \wedge x^{\bot\bot} \leq a$

So the elements above $x$ and $x \wedge x^{\bot\bot}$ are identical; by antisymmetry, $x = x \wedge x^{\bot\bot}$. It follows that $x \leq x^{\bot\bot}.$

A similar argument shows the other direction; therefore $x = x^{\bot\bot}.$

Order-reversingness. The following are equivalent.

1. $x \leq y$
2. $x \wedge 1 \leq y$
3. $1 \leq x^\bot \vee y$
4. $1 \wedge y^\bot \leq x^\bot$
5. $y^\bot \leq x^\bot$

Put $x=a^\perp$, $y=b^\perp$, $z=b$. The first shuffle law fails.
• Yes, they imply orthomodularity and even distributivity, so $L$ must be a Boolean algebra. Indeed, by the first shuffle law, $L$ must be a Heyting algebra/Brouwerian lattice, which is necessarily distributive. Commented Feb 10, 2016 at 23:13