lattice

Lattice

A lattice is an algebraic structure consisting of a set with two binary operations, called meet (denoted $\wedge$) and join (denoted $\vee$), satisfying the following axioms for all elements $a,b,c$ in the set:

commutative: $a\wedge b=b\wedge a$ and $a\vee b=b\vee a$
associativity: $(a\wedge b)\wedge c=a\wedge (b\wedge c)$ and $(a\vee b)\vee c=a\vee (b\vee c)$
absorbtion law: $a\wedge (a\vee b)=a$ and $a\vee (a\wedge b)=a$
idempotence: $a\wedge a=a$ and $a\vee a=a$

EXAMPLE

The power set of a set with intersection ($\cap$, meet) and union ($\cup$, join) forms a lattice, because the power set is closed under these operations and satisfies the lattice axioms.

$\mathbb{N}$ with $\min$ and $\max$ is also a lattice, or with $\gcd$ and $\text{lcm}$.

The set of subspaces of a vector space forms a lattice with intersection and sum. The intersection gives us the largest subspace contained in both (greatest lower bound, just like $\min$ or $\wedge$ or $\gcd$), and the sum gives us the smallest subspace containing both (least upper bound, like $\max$ or $\vee$ or $\text{lcm}$).