From Partial Orders to Lattices

Given two states of knowledge, we can ask: what do they agree on? In the language of partial orders: what is the most informative state that is less precise than both? Only the facts they share survive. This is called the least upper bound (lub), or join, written a ⊔ b.

Now turn it around. Given two separate sets of facts, what does it mean to combine them? In the language of partial orders: what is the least precise state that is more precise than both? We keep everything from both. This is called the greatest lower bound (glb), or meet, written a ⊓ b.

A partial order equipped with both meet and join is called a lattice.

Here is a fragment of the full lattice, built from just three facts — A = "1 at (1,1)", B = "3 at (1,3)", C = "4 at (2,2)". The real lattice is much larger, but this slice is enough to see the structure. Click any two elements to see their meet and join.

But combining facts can produce contradictions — "1 at (1,1)" combined with "2 at (1,1)". We add a special element ⊥ (bottom) to represent inconsistency. It sits below everything.

And we add ⊤ (top) for the state of knowing nothing at all. It sits above everything.