Definition:Lattice (Order Theory)/Definition 2
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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
- Results about lattices in the context of order theory can be found here.
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