Lattice Theory Foundations
Lattice Homomorphisms and Order Preservation
Maps between lattices that respect the operations, and the sharp distinction between lattice homomorphisms and merely order-preserving maps — a distinction that has no analogue in group or ring theory.
Learning objectives
- Define lattice homomorphism, embedding and isomorphism
- Show that every lattice homomorphism is order-preserving
- Produce an order-preserving map that is not a homomorphism
Homomorphisms
A map α: L1 → L2 between lattices is a homomorphism if for all a, b: α(a ∨ b) = α(a) ∨ α(b) and α(a ∧ b) = α(a) ∧ α(b).
A homomorphism preserving only one of the two operations is a join-homomorphism or meet-homomorphism respectively. Both weaker notions are genuinely useful; neither is what “homomorphism” means unqualified.
Homomorphisms preserve order
Every lattice homomorphism is order-preserving: if a ≤ b then α(a) ≤ α(b).
The proof is immediate. Suppose a ≤ b, so a ∨ b = b. Applying α gives α(a) ∨ α(b) = α(b), which says exactly α(a) ≤ α(b).
The converse fails
An order-preserving map between lattices need not be a homomorphism. This is the one place where lattice theory diverges sharply from group and ring theory, where structure-preservation and the natural weak condition coincide.
Counterexample
Take M5, the five-element lattice with three pairwise incomparable middle elements a, b, c between a bottom 0 and a top 1. Map it to the two-element chain {0, 1} by sending 0 and a to 0 and everything else to 1.
This map is order-preserving. But b ∧ c = 0 maps to 0, while α(b) ∧ α(c) = 1 ∧ 1 = 1. Meets are not preserved.
What an order-preserving map does guarantee is one-sided: α(a ∧ b) ≤ α(a) ∧ α(b) and α(a ∨ b) ≥ α(a) ∨ α(b). Equality in both is exactly the homomorphism condition.
Embeddings and isomorphisms
| Map type | Condition | Order behaviour |
|---|---|---|
| Order-preserving | a ≤ b ⇒ α(a) ≤ α(b) | One-directional |
| Order-embedding | a ≤ b ⇔ α(a) ≤ α(b) | Both directions; automatically injective |
| Homomorphism | preserves ∨ and ∧ | Order-preserving |
| Embedding | injective homomorphism | Order-embedding |
| Isomorphism | bijective homomorphism | Order-isomorphism |
For lattices, a bijective order-preserving map whose inverse is also order-preserving is automatically an isomorphism. Suprema and infima are determined by the order, so a bijection respecting order in both directions must respect them.
Frequently asked questions
Why does this distinction not arise for groups?
Because groups carry no independent order to preserve. The phenomenon is specific to structures whose operations are determined by an order relation: the order can be respected weakly without the derived operations being respected at all.
Is an order-embedding always a lattice embedding?
Yes, when the domain is a lattice and the map is an order-embedding onto its image considered as a sub-poset — sup and inf of pairs are then computed identically on both sides. The subtlety is whether the image is closed under the operations of the codomain, which is the sublattice question.
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.2, book pages 10-11.
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.