Definition:Lattice

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Definition

Definition 1

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

Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima.

Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively.


Then the ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice.


Definition 2

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


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

$\left({S, \vee, \preceq}\right)$ is a join semilattice
$\left({S, \wedge, \preceq}\right)$ is a meet semilattice


Definition 3

Let $\left({S, \vee}\right)$ and $\left({S, \wedge}\right)$ be semilattices on a set $S$.

Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$:

$a \vee \left({a \wedge b}\right) = a$
$a \wedge \left({a \vee b}\right) = a$

Let $\preceq$ be the ordering on $S$ defined by:

$\forall a, b \in S: a \preceq b$ if and only if $a \vee b = b$

as on Semilattice Induces Ordering.


Then the ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice.


Also defined as

Some sources refer to a bounded lattice as a lattice.

This comes down to insisting that $\vee$ and $\wedge$ admit identity elements.


Also denoted as

In particular in the context of order theory, it is common to omit $\vee$ and $\wedge$ in the notation for a lattice.

That is, one then writes $\left({S, \preceq}\right)$ in place of $\left({S, \vee, \wedge, \preceq}\right)$.


Also see