Lattices, meet, join, and disjointness

From Posets to Lattices

A lattice is a poset where every pair $(a,b)$ has:

• a meet $a \wedge b$ (greatest lower bound)
• a join $a \vee b$ (least upper bound)

Intuitively:

• $a \wedge b$ is the best common specialization
• $a \vee b$ is the best common generalization

Additional Order Relations

Strict Subsumption

So $a$ is strictly more specific than $b$.

Cover Relation

Element $a$ covers $b$ (often written $b \prec a$) when:

• $b \sqsubset a$, and
• there is no $c$ with $b \sqsubset c \sqsubset a$

Cover relations define the edges in a Hasse diagram.

Overlap and Disjointness

Overlap

$a$ and $b$ overlap if they have a common lower bound that is not impossible/trivial:

Disjointness

In bounded settings, a common criterion is:

meaning they share no nontrivial specialization.

Practical Reading

• Use join to compute a common abstraction.
• Use meet to test compatibility.
• Use disjointness to detect impossible intersections.

Terminology Note

In order theory, “join” means supremum ($\vee$). In monad theory, “join” means flattening $M(MA) \to MA$. They are different operations that share a name.

See:

• Order Theory
• Specialization Orders and Subsumption
• Introduction to Categories
• Monads and Applicatives