Definition:Lattice (Order Theory)

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This page is about Lattice in the context of Order Theory. For other uses, see Lattice.

Definition

Definition 1

Let $\struct {S, \preceq}$ 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 $\struct {S, \vee, \wedge, \preceq}$ is called a lattice.


Definition 2

Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.


Then $\struct {S, \vee, \wedge, \preceq}$ is called a lattice if and only if:

$(1): \quad \struct {S, \vee, \preceq}$ is a join semilattice

and:

$(2): \quad \struct {S, \wedge, \preceq}$ is a meet semilattice.


Definition 3

Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$.

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

$a \vee \paren {a \wedge b} = a$
$a \wedge \paren {a \vee b} = a$

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 $\struct {S, \vee, \wedge, \preceq}$ 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 the particular context of order theory, it is common to omit $\vee$ and $\wedge$ in the notation for a lattice.

That is, one then writes $\struct {S, \preceq}$ in place of $\struct {S, \vee, \wedge, \preceq}$.


Also known as

Some sources, in order to reduce confusion between a lattice and a point lattice, use the term order lattice for the former.


Also see

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