Lattice Theory Foundations
Posets and the Two Definitions of a Lattice
A lattice is either an ordered set in which every pair has a supremum and an infimum, or an algebra with two idempotent commutative associative operations linked by absorption. Proving these are the same object is the first real theorem of the subject.
- Define a lattice as a poset and as an algebra of type (2, 2).
- State the four pairs of identities that axiomatise a lattice algebraically.
- Recover the order relation from the operations and vice versa.
- Explain why the algebraic definition is the one universal algebra requires.
- Identify bounded, complemented and lattice-ordered variants correctly.
01The order-theoretic definition
Start with a partially ordered set: a set with a reflexive, antisymmetric, transitive relation ≤. For a subset X, an upper bound is an element above everything in X, and the least upper bound — the supremum or join — is the smallest such element when one exists. Greatest lower bound, infimum and meet are dual.
A poset is a lattice when every two-element subset has both a least upper bound and a greatest lower bound. By induction this gives sups and infs for all finite non-empty subsets, but not for infinite ones — that stronger condition defines a complete lattice and is a genuinely different notion.
Suprema and infima are unique when they exist, which is what licenses writing them as operations. Uniqueness follows from antisymmetry: two least upper bounds are each below the other, hence equal. Without antisymmetry — in a preordered set — the construction fails and no algebra results.
02The algebraic definition
The alternative definition discards the order entirely and posits two binary operations satisfying eight identities in four dual pairs.
| Pair | Join form | Meet form |
|---|---|---|
| Idempotency | x ∨ x ≈ x | x ∧ x ≈ x |
| Commutativity | x ∨ y ≈ y ∨ x | x ∧ y ≈ y ∧ x |
| Associativity | x ∨ (y ∨ z) ≈ (x ∨ y) ∨ z | x ∧ (y ∧ z) ≈ (x ∧ y) ∧ z |
| Absorption | x ∨ (x ∧ y) ≈ x | x ∧ (x ∨ y) ≈ x |
Absorption is the pair that does the real work. Without it one has two independent semilattices with no relationship; absorption is precisely the condition that forces them to induce the same order. Note also that idempotency is derivable from absorption together with the rest, so the axiom set is not independent — it is presented redundantly because the dual symmetry is more useful than minimality.
03Proving the two definitions agree
- given a lattice poset ⟨L, ≤⟩:
- define x ∨ y := sup{x, y} and x ∧ y := inf{x, y}
- verify the eight identities directly from the sup/inf characterisation
- given a lattice algebra ⟨L, ∨, ∧⟩:
- define x ≤ y iff x ∨ y = y
- (equivalently, by absorption, x ≤ y iff x ∧ y = x)
- verify ≤ is reflexive (idempotency), antisymmetric (commutativity),
- and transitive (associativity)
- verify x ∨ y = sup{x, y} and x ∧ y = inf{x, y} under ≤
- the two translations are mutually inverse
The step worth dwelling on is the equivalence of the two definitions of the order. If x ∨ y = y then x ∧ y = x ∧ (x ∨ y) = x by absorption; the converse is symmetric. Every subsequent argument that slides between the two forms is using this.
04Why universal algebra insists on the algebraic form
Both definitions describe the same objects, so the choice might look like taste. It is not. The HSP theorem says that a class is a variety exactly when it is equationally definable, and only the algebraic definition is equational.
A subset of a lattice that happens to be a lattice under the inherited order need not be a sublattice, because its sups and infs may differ from those computed in the parent. A sublattice must be closed under the parent's ∨ and ∧. This distinction is a recurring source of error, and it is exactly the distinction the algebraic definition makes unmissable.
05Bounded, complemented and related variants
Several standard enrichments add operations to the type, and each changes the variety and its subalgebra lattice.
| Type | Operations | Consequence |
|---|---|---|
| Lattice | (2, 2) | The base case. May have no greatest or least element. |
| Bounded lattice | (2, 2, 0, 0) | Constants 0 and 1 in the type, so every subalgebra contains them. |
| Complemented lattice | (2, 2, 0, 0) + axiom | Complements need not be unique unless the lattice is distributive. |
| Boolean algebra | (2, 2, 1, 0, 0) | Complementation is an operation, hence preserved by homomorphisms. |
| Lattice-ordered group | (2, 2, 2, 1, 0) | Group and lattice structure interacting; a variety in the combined type. |
The Boolean case illustrates the general point. Making complementation a unary operation rather than an existence axiom means homomorphisms must preserve it, which is why Boolean algebra homomorphisms are much more rigid than lattice homomorphisms between the same objects.
Frequently asked
Is every partially ordered set a lattice?
No. The poset consisting of two incomparable elements has no upper bound for the pair at all, so no least upper bound. Even a poset with all upper bounds present may fail: if two elements have several minimal upper bounds and none is below the others, there is no supremum. Failure of sup or inf on a single pair is enough to disqualify the poset.
Are the eight lattice identities independent?
No. Idempotency follows from absorption and the remaining axioms: x ∨ x = x ∨ (x ∧ (x ∨ x)) = x by two applications of absorption. The redundant presentation is retained because the dual symmetry between the join and meet columns makes every subsequent proof come in pairs, and halving the axioms would destroy that.
Why can't completeness be added as an identity?
Because identities are equations between terms, and terms are finite. An identity can only ever constrain finitely many elements at once. Completeness asserts the existence of suprema for arbitrary subsets, including infinite ones, which no finite term can express. This is why complete lattices do not form a variety and why the class is not closed under the usual constructions.
- S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
- G. Grätzer, Universal Algebra, 2nd edition, Springer.
- R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.
Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.