Lattice Theory Foundations
Lattices as Algebras: the Equational Definition
A lattice can be defined purely equationally, as a set with two binary operations satisfying four pairs of identities. This is the definition that makes lattices algebras in the sense of universal algebra, and it is the one the subject uses.
Learning objectives
- State the four defining identity pairs L1–L4
- Verify that a candidate structure is a lattice from the identities alone
- Explain why the equational definition matters for universal algebra
The four identity pairs
A lattice is a non-empty set L together with two binary operations ∨ and ∧ satisfying, identically in L:
| Label | Identity (a) | Identity (b) | Name |
|---|---|---|---|
| L1 | x ∨ y ≈ y ∨ x | x ∧ y ≈ y ∧ x | commutative laws |
| L2 | x ∨ (y ∨ z) ≈ (x ∨ y) ∨ z | x ∧ (y ∧ z) ≈ (x ∧ y) ∧ z | associative laws |
| L3 | x ∨ x ≈ x | x ∧ x ≈ x | idempotent laws |
| L4 | x ≈ x ∨ (x ∧ y) | x ≈ x ∧ (x ∨ y) | absorption laws |
L1–L3 say each operation makes L a commutative idempotent semigroup. It is L4 that ties the two operations together. Without absorption you have two unrelated semilattice structures on the same set; with it, each determines the other.
Two worked examples
Propositions
Let L be the set of propositions, with ∨ reading as “or” and ∧ as “and”. L1–L4 are then familiar facts of propositional logic. Absorption reads: p is equivalent to p or (p and q).
Natural numbers under lcm and gcd
Let L be the natural numbers, ∨ the least common multiple and ∧ the greatest common divisor. Each identity is a routine fact of elementary number theory. Absorption reads: lcm(a, gcd(a, b)) = a.
These two examples are worth holding onto, because they behave differently later: the propositional lattice is distributive and complemented, while the divisibility lattice is distributive but not complemented.
Why the equational form matters
Universal algebra's central theorem — Birkhoff's HSP theorem — applies exactly to classes defined by identities. Because lattices are defined by identities, the class of all lattices is a variety, and everything the general theory proves about varieties applies to it immediately.
- The class of lattices is closed under homomorphic images, subalgebras and direct products.
- Free lattices exist on every generating set.
- Every lattice is a subdirect product of subdirectly irreducible lattices.
- Lattice identities can be derived by the formal rules of equational logic.
None of this would follow from the order-theoretic definition on its own. That definition is often more convenient for calculation, but it is the equational one that connects lattices to the rest of the subject.
Duality
The identity list is symmetric: interchanging ∨ and ∧ throughout maps L1(a) to L1(b), L2(a) to L2(b), and so on. The system is therefore self-dual.
If a statement expressible in terms of ∨ and ∧ holds in all lattices, then so does the statement obtained by interchanging ∨ and ∧ throughout. Every theorem comes free with its dual.
This halves the work in Chapter I and continues to pay off in Chapter IV, where Boolean algebras carry a stronger duality of their own.
Frequently asked questions
Do I need to assume a partial order to define a lattice?
No — that is the point of the equational definition. The order is recovered from the operations, not assumed alongside them. The two definitions turn out to be equivalent, which is the content of the equivalence theorem.
Are the four pairs independent?
Idempotence is in fact derivable from absorption: applying L4(b) then L4(a) yields x ∨ x ≈ x. The list is stated redundantly for clarity rather than minimality.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section I.1, book pages 5-6.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.