I'm just starting out looking at lattices, and I've hit a bump. I'm using Applied Abstract Algebra by Lidl and Pilz. On p. 7, the authors define the duality principle:
Any "formula" in a lattice $ ( L, \sqcap, \sqcup ) $ involving the operations $ \sqcap $ and $ \sqcup $ remains valid if we replace $ \sqcap $ by $ \sqcup $ and $ \sqcup $ by $ \sqcap $ everywhere in the formula.
So far, so good. On p. 10, then, we see that
[i]n every lattice $ L $ the operations $\sqcap $ and $\sqcup$ are isotone, i.e. $y\le z \implies x \sqcap y \le x \sqcap z$ and $x \sqcup y \le x \sqcup z$.
I think that the proof seems straightforward enough:
$ y \le z \implies x \sqcap y = (x \sqcap x) \sqcap (y \sqcap z) = (x \sqcap y) \sqcap (x \sqcap z) $
— whence the first statement. (The last equality is obtained by applying associativity and commutativity, I assume?)
The authors then write
The second formula is verified by duality.
But $y \le z$ does not imply that $ x \sqcup y = (x \sqcup x) \sqcup (y \sqcup z)$; rather, we would need the LHS to be $x \sqcup z$. I thought perhaps this was simply a matter of less-than-precise language (yes, I realize that's very unlikely), that perhaps I needed to make this simple change and then everything works out. But there's nothing in the definition of the duality principle on p. 7 about making other "simple changes". What am I missing?