Definition:Lattice

Definition

Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ is a lattice iff:

$\forall x, y \in S: \left\{{x, y}\right\} \subseteq S$ admits both a supremum and an infimum.

In the context of a lattice, the following symbols are often seen:

• $\sup \left\{{x, y}\right\}$ is written as $x \cup y$ or $x \vee y$ and is called the union of $x$ and $y$

• $\inf \left\{{x, y}\right\}$ is written as $x \cap y$ or $x \wedge y$ and is called the intersection of $x$ and $y$.

Class

Let $C$ be a class.

Let $\thickapprox$ be an equivalence relation on $C$.

Let $\left({C, \vee}\right)$ and $\left({C, \wedge}\right)$ be a semilattices with respect to $\thickapprox$.

Then $\left({C, \vee, \wedge}\right)$ is a lattice with respect to $\thickapprox$ iff:

$\forall x, y \in C: $

$x \vee (x \wedge y) \thickapprox x$, and

$x \wedge (x \vee y) \thickapprox x$.

These identities are referred to as the absorption laws.

If $\left({S, \preceq}\right)$ is a lattice by definition 1, then $\left({S, \vee, \wedge}\right)$ is a lattice with respect to $=$ by definition 2 with $\vee$ and $\wedge$ defined as in definition 1.