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.

Examples

Divisor Relation

Let $\struct {\Z_{>0}, \divides}$ denote the order structure consisting of the (strictly) positive integers under the divisor relation $\divides$.

Then $\struct {\Z_{>0}, \divides}$ is a lattice.

Also see

• Definition:Lattice (Order Theory): an abstraction of this definition in a more general context

Historical Note

The concept of a lattice was initiated by Richard Dedekind in $1894$.

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): $\S 7$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): lattice: 1. (in algebra; R. Dedekind, 1894)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): lattice: 1. (in algebra; R. Dedekind, 1894)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order