Definition:Lattice (Order Theory)
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
- 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 in the context of order theory can be found here.