Definition:Lattice (Ordered Set)

Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

for all $x, y \in S$, the subset $\set {x, y}$ admits both a supremum and an infimum.

Also see

• Definition:Lattice: an abstraction of this definition in a more general context

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$