Lattice Homomorphisms, Isomorphisms and Sublattices

01Homomorphisms preserve operations, not merely order

A lattice homomorphism is a map α with α(x ∨ y) = α(x) ∨ α(y) and α(x ∧ y) = α(x) ∧ α(y). Every such map is monotone: if x ≤ y then x ∨ y = y, so α(x) ∨ α(y) = α(y), giving α(x) ≤ α(y).

The failure is instructive: monotone maps preserve the order relation but need not preserve leastness of upper bounds. Preservation of the operations is a strictly stronger requirement, and it is the requirement that makes lattices algebras in the universal-algebraic sense.

02Isomorphism: where order and algebra coincide

For isomorphisms the gap closes. A bijection between lattices is a lattice isomorphism if and only if both it and its inverse are monotone — that is, if and only if it is an order isomorphism.

This is a genuine peculiarity of lattices. In groups a bijective homomorphism is automatically an isomorphism; in lattices a bijective monotone map is not, though a bijective homomorphism still is. The distinction is worth keeping straight because order-theoretic intuition suggests otherwise.

03Sublattices and the closure requirement

A sublattice is a subset closed under the parent lattice's join and meet. The standard trap is a subset that forms a lattice under the inherited order while computing different sups and infs than the parent does.

1. Take the subset and the inherited order
   Check whether every pair within the subset has a sup and inf within the subset. If not, it is not even a sub-poset lattice.
2. Compare against the parent's operations
   For each pair, compute x ∨ y and x ∧ y in the parent. If either falls outside the subset, the subset is not a sublattice however well behaved it looks internally.
3. Conclude
   Closure under the parent operations is the definition. Nothing weaker suffices, because homomorphic image and product arguments all use the parent operations.

Because sublattices are exactly the subuniverses of the algebra ⟨L, ∨, ∧⟩, they form an algebraic lattice under inclusion — a fact that generalises to every algebra and is developed in the Core stream.

04Embeddings and forbidden configurations

An embedding is an injective homomorphism, so its image is a sublattice isomorphic to the source. Embeddings matter because the central classification results in elementary lattice theory are stated as forbidden sublattices.

05Congruences on lattices, briefly

Lattice congruences behave less conveniently than group congruences, and it is worth seeing why early. Lattices are congruence-distributive, which is a strong and useful property, but they are not congruence-permutable.

The consequence for practice is that computing a join of two lattice congruences is genuinely harder than computing a meet, and arguments that implicitly assume θ ∨ φ = θ ∘ φ are simply invalid here.