Lattice - Knowlpedia

Definition

A lattice is a partially ordered set $(L,\leq)$ in which every pair $a,b\in L$ has both a least upper bound, called its join $a\vee b$ , and a greatest lower bound, called its meet $a\wedge b$ .

Algebraic laws

Join and meet are commutative, associative, and idempotent, and they satisfy the absorption laws

a\vee(a\wedge b)=a,\qquad a\wedge(a\vee b)=a.

Conversely, two binary operations satisfying these laws determine a partial order by

a\leq b\quad\Longleftrightarrow\quad a\vee b=b \quad\Longleftrightarrow\quad a\wedge b=a.

Thus order-theoretic and algebraic definitions of a lattice are equivalent.

Examples and scope

The subsets of a set form a lattice under inclusion, with union as join and intersection as meet. Every total order is a lattice, with maximum as join and minimum as meet, but a lattice need not be totally ordered.

A lattice homomorphism preserves binary joins and binary meets. A bounded lattice additionally has a least and a greatest element; a homomorphism of bounded lattices is normally also required to preserve those bounds. Boundedness is not part of the definition here.

References

Garrett Birkhoff, Lattice Theory, 3rd ed., American Mathematical Society, 1967. AMS record. Relevant: Chapter I.