Lattices

What this lesson covers

This lesson introduces lattices: partially ordered sets where every pair of elements has a greatest lower bound (meet) and a least upper bound (join). By the end, you will be able to determine whether a poset is a lattice, compute meets and joins, and recognize bounded and complete lattices.

Why it matters

A partial order lets you say “this is below that,” but it does not guarantee you can combine two elements. Given two topics in a library — say Modal Logic and Set Theory — a partial order under “is more specific than” tells you they are both below Mathematics. But is there a most specific topic that encompasses both? Is there a most general topic that both encompass?

A lattice guarantees these exist. For every pair of elements, there is a best common upper bound (their join) and a best common lower bound (their meet). This means you can always combine and intersect — you can always ask “what do these have in common?” (meet) and “what covers both of these?” (join).

Lattices are the stepping stone to Heyting Algebras, which add a notion of implication — and thus a logic — to the lattice structure. Without meets and joins, there is no implication. Without a lattice, there is no Heyting algebra. The entire chain from partial orders through closure operators depends on this structure.

Prerequisites

• Partial Orders: the three axioms (reflexive, antisymmetric, transitive), Hasse diagrams, comparability and incomparability. You should be comfortable checking whether a relation is a partial order and drawing its Hasse diagram.

Core concepts

Bounds

Given elements $a$ and $b$ in a poset $(S,\le )$:

• An element $\ell$ is a lower bound of $\{a,b\}$ if $\ell \le a$ and $\ell \le b$.
• An element $u$ is an upper bound of $\{a,b\}$ if $a\le u$ and $b\le u$.

There may be many lower bounds and many upper bounds — or none at all.

Example. In the divisibility order on $\{1,2,3,6\}$: the lower bounds of $\{2,3\}$ are just $\{1\}$ (since $1$ divides both $2$ and $3$). The upper bounds of $\{2,3\}$ are just $\{6\}$ (since both $2$ and $3$ divide $6$).

Meet and join

The meet of $a$ and $b$, written $a\wedge b$, is the greatest lower bound — the lower bound that is above all other lower bounds. Formally, $a\wedge b=\ell$ where:

• $\ell \le a$ and $\ell \le b$ (it is a lower bound), and
• for every lower bound ${\ell }^{\prime }$ of $\{a,b\}$, ${\ell }^{\prime }\le \ell$ (it is the greatest such).

The join of $a$ and $b$, written $a\vee b$, is the least upper bound — the upper bound that is below all other upper bounds.

If the meet or join does not exist for some pair, the poset is not a lattice.

Lattice

A lattice is a poset where every pair of elements has a meet and a join. Equivalently, the operations $\wedge$ and $\vee$ are defined for all pairs and satisfy:

• Commutative: $a\wedge b=b\wedge a$ and $a\vee b=b\vee a$.
• Associative: $(a\wedge b)\wedge c=a\wedge (b\wedge c)$ and likewise for $\vee$.
• Absorption: $a\wedge (a\vee b)=a$ and $a\vee (a\wedge b)=a$.
• Idempotent: $a\wedge a=a$ and $a\vee a=a$.

These are not additional axioms — they follow from the definition of meet and join as greatest lower bound and least upper bound.

Bounded lattice

A bounded lattice is a lattice with a greatest element $\top$ (top) and a least element $\perp$ (bottom), satisfying:

• $\perp \le a\le \top$ for every $a$.
• $a\wedge \top =a$ and $a\vee \perp =a$ for every $a$.

Most lattices encountered in practice are bounded. The existence of $\top$ and $\perp$ is required for defining Heyting implication in the next lesson.

Complete lattice

A complete lattice is a lattice where every subset (not just every pair) has a meet and a join. This is stronger than being a lattice: it means you can take the meet of infinitely many elements and get a result.

Every finite lattice is automatically complete (since you can compute the meet or join of any subset by iterating pairwise operations). Infinite lattices need not be complete.

Worked example: the power set lattice

Continuing from the previous lesson, let $X=\{a,b\}$ and consider $(\mathcal{P}(X),\subseteq )$ — the subsets of $X$ ordered by inclusion.

Computing meets and joins:

Pair	Meet ($\wedge$ = $\cap$)	Join ($\vee$ = $\cup$)
$\{a\},\{b\}$	$\varnothing$	$\{a,b\}$
$\{a\},\{a,b\}$	$\{a\}$	$\{a,b\}$
$\varnothing ,\{b\}$	$\varnothing$	$\{b\}$

Every pair has a meet and a join, so this is a lattice.

It is bounded: $\perp =\varnothing$ and $\top =\{a,b\}$.

It is complete: the meet of any collection of subsets is their intersection, and the join is their union. (This works for arbitrary collections, not just pairs.)

Hasse diagram (same as before, now annotated with lattice operations):

   {a,b} = ⊤
   /   \
 {a}   {b}      {a} ∧ {b} = ∅,  {a} ∨ {b} = {a,b}
   \   /
    ∅ = ⊥

This lattice will serve as the basis for the Heyting algebra in the next lesson, where we add an implication operation.

A non-example: a poset that is not a lattice

Consider $S=\{a,b,c,d\}$ with:

  c   d
  |   |
  a   b

where $a\le c$, $b\le d$, and no other non-trivial comparisons hold. The pair $\{a,b\}$ has no upper bound — neither $c$ nor $d$ is above both $a$ and $b$. Since the join of $\{a,b\}$ does not exist, this is not a lattice.

Adding a top element $\top$ above both $c$ and $d$ would make the join of $\{a,b\}$ exist (it would be $\top$), but you would also need a bottom element $\perp$ below both $a$ and $b$ for the meet. The structure needed for a lattice is tighter than for a mere poset.

Check your understanding

1. In the divisibility order on $\{1, 2, 3, 6\}$, what is $2 \wedge 3$? What is $2 \vee 3$?

$2\wedge 3=1$ (the greatest common divisor). $2\vee 3=6$ (the least common multiple). Since meets and joins exist for all pairs, $(\{1,2,3,6\},|)$ is a lattice. It is bounded with $\perp =1$ and $\top =6$.

2. Is the divisibility order on $\{2, 3, 4, 6\}$ a lattice?

Consider $4$ and $6$. Their lower bounds are the elements dividing both: $2$ divides both, so $2$ is a lower bound. Is $2$ the greatest lower bound? There is no element $>2$ that divides both $4$ and $6$ ($3$ does not divide $4$, $4$ does not divide $6$). So $4\wedge 6=2$. Now check the join: $4\vee 6$ would be the least common multiple of $4$ and $6$, which is $12$. But $12\notin S$. Since the join of $\{4,6\}$ does not exist in $S$, this is not a lattice.

3. Why can't you define [[Heyting Implication|Heyting implication]] without meets and joins?

Heyting implication $a\Rightarrow b$ is defined as the greatest element $c$ such that $c\wedge a\le b$. Computing this requires that meets exist (to make sense of $c\wedge a$) and that the set of all such $c$ has a greatest element (which is guaranteed in a lattice but not in a general poset). Without the lattice structure, implication cannot be defined.

Common mistakes

• Confusing “the meet exists” with “a lower bound exists.” A lower bound of $\{a,b\}$ is any element below both. The meet is the greatest such element. There may be many lower bounds but no greatest one (in which case the meet does not exist and the poset is not a lattice).
• Assuming all posets are lattices. Many natural orders (like the divisibility order on arbitrary finite sets) are not lattices. Always check that meets and joins exist for every pair.
• Forgetting boundedness. A lattice need not have a top and bottom. But Heyting algebras and most structures in this sequence require boundedness. When moving to the next lesson, verify that $\top$ and $\perp$ exist.

What comes next

The next lesson, Heyting Algebras, adds an implication operation to the bounded lattice. Given the lattice structure, $a\Rightarrow b$ is defined as the greatest element whose meet with $a$ stays below $b$. This gives the lattice the structure of an intuitionistic logic — where “or” is not always decidable and double negation does not always cancel. Heyting algebras are the algebraic foundation of the Semiotic Universe and the structures that emerge from the relational derivation.