Lattice Theory Foundations
Sublattices and Lattice Isomorphism
Subsets of a lattice that are lattices in their own right under the inherited operations, the difference between a sublattice and a sub-poset that happens to be a lattice, and the classification of lattices up to isomorphism.
Learning objectives
- Define sublattice and test whether a subset qualifies
- Distinguish a sublattice from a sub-poset that is independently a lattice
- Recognise isomorphic lattices from their diagrams
Sublattices
A non-empty subset L′ of a lattice L is a sublattice if it is closed under both operations: whenever a, b lie in L′, so do a ∨ b and a ∧ b, computed in L.
The emphasis on “computed in L” is the whole content of the definition. A sublattice inherits the ambient operations; it does not get to recompute them.
The trap: sub-posets that are lattices but not sublattices
“Is this subset closed under the ambient ∨ and ∧?” and “Is this subset, with the inherited order, a lattice?” are different questions with different answers.
Illustration
Take N5: bottom 0, top 1, a chain a < b on one side, and a single element c on the other, with c incomparable to both a and b.
The subset {0, a, c, 1} is a sub-poset that is a lattice in its own right. But in the ambient N5 the join a ∨ c equals 1, and this happens to lie in the subset — so here the subset is a sublattice. Subsets where the ambient join or meet escapes the subset fail the test; a subset may still be a lattice under its own inherited order because the sup taken within the subset differs from the sup taken in L.
The practical test is mechanical: for every pair in the candidate subset, compute the join and the meet in the ambient lattice and check membership. Nothing else counts.
Standard sublattice constructions
Intervals
For a ≤ b, the interval [a, b] = {x : a ≤ x ≤ b} is always a sublattice.
Principal ideals
The down-set below a fixed element is a sublattice, and is closed downward.
Principal filters
The up-set above a fixed element is a sublattice, and is closed upward.
Intersections
Any intersection of sublattices is a sublattice, which is why the sublattices of L form a closure system.
That intersections of sublattices are sublattices is the lattice case of a fact holding for every algebra: intersections of subuniverses are subuniverses. That fact is what makes Sub(A) a complete lattice, and it is proved in general in Chapter II §3.
Isomorphism
Lattices L1 and L2 are isomorphic if there is a bijection between them preserving both operations. Equivalently, there is an order-isomorphism between them.
For finite lattices, isomorphism is decided by comparing Hasse diagrams: two finite lattices are isomorphic exactly when their diagrams can be redrawn to coincide. This is why the small lattices M5 and N5 can be characterised by picture alone.
| Size | Count | Notes |
|---|---|---|
| 1 | 1 | Trivial lattice |
| 2 | 1 | The two-element chain 2 |
| 3 | 1 | The three-element chain |
| 4 | 2 | The four-chain and the “diamond” 2×2 |
| 5 | 5 | Includes M5 and N5 |
Frequently asked questions
Is a subset closed under meet but not join still useful?
Yes — it is a meet-subsemilattice, and such subsets appear naturally. But it is not a sublattice, and results about sublattices do not apply to it.
Why do M5 and N5 matter so much?
Because they characterise modularity and distributivity by exclusion: a lattice is modular exactly when N5 does not embed as a sublattice, and distributive exactly when neither M5 nor N5 does. Those two theorems are the subject of a separate page.
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.