Definition:Lattice

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

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:

$\struct {S, \vee, \preceq}$ is a join semilattice

$\struct {S, \wedge, \preceq}$ 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 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 $\struct {S, \preceq}$ in place of $\struct {S, \vee, \wedge, \preceq}$.

Also see

• Equivalence of Definitions of Lattice

• Definition:Bounded Lattice
• Definition:Join Semilattice
• Definition:Meet Semilattice
• Definition:Semilattice
• Definition:Complete Lattice
• Definition:Lattice (Ordered Set): a specific instantiation of a lattice in the context of an ordered set

• Results about lattices can be found here.

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