I'm studying distributivity in the lattice theory, consequently, I'm after any ideas that might help to develop some intuition.

In elementary school level algebra, distributivity property

```
(a+b) c = ac + bc
```

is pictured geometrically as the two adjacent rectangles: `a x c`

and `b x c`

. (Geometric interpretation follows from linearity of inner product space, I guess).

Next, for Boolean algebras (which are lattices) there are the two interpretations: Venn diagram, and cube.

On Venn diagram we draw union of sets a and b, then intersect the result with c. Then, we compare it with a intersect with c unioned with b intersect with c. One can convince yourself that the result is the same, although it is less intuitive than in the previous case of adjacent rectangles.

When I do the same "operational" drawing on (4 dimensional) Boolean cube image, the result is again the same, but the procedure is even less convincing. Why `(a v b) ^ c`

is the same as `(a ^ c) v (b ^ c)`

, is it due to some properties of the geometric shape of the cube?

In a similar fashion I drew couple of cases on graph paper (representing N^2 lattice). Again, why `(a v b) ^ c`

is the same as `(a ^ c) v (b ^ c)`

is not obvious at all.

Both Boolean cube and N^2 lattice are Cartesian products of 1-dimensional lattices which are total orders. IIRC, distributivity carries over to Cartesian products, so one just have to analyze 1-dimensional picture only. So we have to see that

```
min(max(a,b),c) = max(min(c,a),min(c,b))
```

holds in every of the following cases:

```
a < b < c
a < c < b
c < a < b
```

(where transpositions of a and b are ignored because of commutativity)

Again, this case analysis is not exactly geometrically evident interpretation...

Group theory promoted non-commutativity property into a whole new topic; by analogy, had anybody try to introduce "distributor", that is `[a,b,c]`

(this is reminiscent of standard group commutator notation) such that

```
(a v b) ^ c = (a ^ c) v (b ^ c) v [a,b,c]
```

or there is a reason why this wouldn't work?