Lattices

Abbreviation: Lat

Definition

A lattice is a structure L = (L,∨,∧), where ∨ and ∧ are infix binary operations called the join and meet, such that
∨,∧ are associative:   (x∨y)∨z = x∨(y∨z), (x∧y)∧z = x∧(y∧z)
∨,∧ are commutative:   x∨y = y∨x, x∧y = y∧x, and
∨,∧ are absorbtive:   (x∨y)∧x = x, (x∧y)∨x = x.

Remark: It follows that ∨ and ∧ are idempotent: x∨x = x, x∧x = x.
This definition shows that lattices form a variety.
A partial order  ≤  is definable in any lattice by x ≤ y  ⇔   x∧y = x, or equivalently by x ≤ y  ⇔   x∨y = y.

Morphisms

Let L and M be lattices. A morphism from L to M is a function h : L→M that is a homomorphism: h(x∨y) = h(x)∨h(y)  and  h(x∧y) = h(x)∧h(y).

Definition

A lattice is a structure L = (L,∨,∧) of type (2,2) such that
(L,∨) and (L,∧) are semilattices, and
∨,∧ are absorbtive:   (x∨y)∧x = x, (x∧y)∨x = x.

Definition

A lattice is a structure L = (L, ≤ ) that is a poset in which all elements x,y ∈ L have a
least upper bound:   z = x∨y  ⇔   x ≤ z  and  y ≤ z  and   ∀w (x ≤ w  and  y ≤ w  ⇒  z ≤ w), and
greatest lower bound:   z = x∧y  ⇔   z ≤ x  and  z ≤ y  and   ∀w (w ≤ x  and  w ≤ y  ⇒  w ≤ z).

Definition

A lattice is a structure L = (L,∨,∧, ≤ ) such that (L, ≤ ) is a poset and the following quasiequations hold:
∨-left:   x ≤ z  and  y ≤ z   ⇒  x∨y ≤ z,
∨-right:   z ≤ x  ⇒  z ≤ x∨y,     z ≤ y  ⇒  z ≤ x∨y,
∧-right:   z ≤ x  and  z ≤ y  ⇒  z ≤ x∧y,
∧-left:   x ≤ z  ⇒  x∧y ≤ z,     y ≤ z  ⇒  x∧y ≤ z.

Remark: These quasiequations give a cut-free Gentzen system to decide the equational theory of lattices.

Some results

Examples

(P(S),∪,∩, ⊆ ), the collection of subsets of a sets S, ordered by inclusion.

Properties

Finite members

[Jobst Heitzig, Jürgen Reinhold, Counting finite lattices, Algebra Universalis 48 (2002) 43--53 MRreview]

[Lattices of size 1 to 6]
Finite lattices of size  ≤ 7
[Subdirectly irreducible lattices]? of size  ≤ 7
[Lattices not in some subclasses]? of size  ≤ 7

Subclasses

Superclasses