Definition:Lattice (Order Theory)/Definition 2

Definition

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.

That is, for all $a, b \in S$:

$a \vee b$ is the supremum of $\set {a, b}$

and:

$a \wedge b$ is the infimum of $\set {a, b}$

Also see

• Equivalence of Definitions of Lattice (Order Theory)

• Results about lattices in the context of order theory can be found here.

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