Lattice Theory Foundations
Lattices as Posets and the Equivalence Theorem
The order-theoretic definition of a lattice and the theorem establishing that it agrees exactly with the equational definition. Both directions of the construction are given, together with the reason the correspondence is a genuine equivalence.
Learning objectives
- State the order-theoretic definition of a lattice
- Recover the order from the operations and the operations from the order
- Explain why the two constructions are mutually inverse
The order-theoretic definition
A poset L is a lattice if for every pair of elements a, b in L both sup{a, b} and inf{a, b} exist in L.
Given such a poset one defines operations by a ∨ b = sup{a, b} and a ∧ b = inf{a, b}.
From operations to order
Conversely, given a lattice in the equational sense, define
a ≤ b if and only if a ∨ b = b.
Absorption gives the equivalent formulation a ≤ b if and only if a ∧ b = a; the two agree in every lattice. That this relation is a partial order follows from L1–L4:
The equivalence theorem
The two constructions are mutually inverse:
- Starting from a lattice in the equational sense, the induced relation ≤ is a partial order under which sup and inf of each pair exist and agree with ∨ and ∧.
- Starting from a lattice in the order sense, the operations sup and inf satisfy L1–L4, and the order they induce is the original order.
The two definitions therefore describe the same objects, and the subject moves between them without comment. Notation reflects this: ≤ is used freely in an algebraically defined lattice, and ∨, ∧ in an order-theoretically defined one.
Use the equational definition when applying general universal-algebraic machinery — varieties, free algebras, congruences. Use the order definition when computing, drawing diagrams, or reasoning about completeness. Neither is primary.
Consequences of the correspondence
- Monotonicity. Both operations are order-preserving in each argument: a ≤ b implies a ∨ c ≤ b ∨ c and a ∧ c ≤ b ∧ c.
- Bounds within the lattice. a ∧ b ≤ a ≤ a ∨ b always.
- Finite joins and meets. Induction on L2 extends both operations to any finite non-empty subset. Infinite subsets need completeness, which is a genuinely stronger condition.
- Duality made concrete. Reversing the order of a lattice yields another lattice, the dual, with ∨ and ∧ interchanged.
The correspondence is what lets Con(A) be treated as both an algebraic object and an ordered one. Congruences are ordered by inclusion — a purely order-theoretic fact — while joins and meets of congruences are computed algebraically. Every later result about congruence lattices trades on both views at once.
Frequently asked questions
Is the induced order the only one compatible with the operations?
Yes. If a partial order induces the given join and meet as sup and inf, then a ≤ b must be equivalent to a ∨ b = b, so the order is determined.
Do lattices need top and bottom elements?
Not in general. A lattice has all finite non-empty joins and meets, but need not have a greatest or least element — the integers under min and max form a lattice with neither. Bounded lattices are those that do.
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 8-9.
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.