3.2 Intervals in lattices

Intervals in lattices are subsets that include all elements between two given points. They come in different types: closed, open, and half-open. Understanding intervals helps us grasp the structure and relationships within lattices.

Intervals play a crucial role in analyzing lattice properties. They can form sublattices and lattice segments, which are important for studying bounded lattices and convex subsets. These concepts are fundamental for exploring more complex lattice structures.

Interval Types

Defining and Classifying Intervals

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• An interval in a lattice $L$ is a subset $I \subseteq L$ such that if $x, y \in I$ and $x \leq z \leq y$, then $z \in I$
• Intervals are classified based on whether their endpoints are included in the set
• A closed interval $[a, b]$ includes both endpoints $a$ and $b$ ($a \leq x \leq b$ for all $x \in [a, b]$)
• An open interval $(a, b)$ excludes both endpoints $a$ and $b$ ($a < x < b$ for all $x \in (a, b)$)
• A half-open interval includes only one of its endpoints, either $[a, b)$ or $(a, b]$
  • $[a, b)$ includes $a$ but excludes $b$ ($a \leq x < b$ for all $x \in [a, b)$)
  • $(a, b]$ excludes $a$ but includes $b$ ($a < x \leq b$ for all $x \in (a, b]$)

Properties and Examples of Intervals

• Intervals are convex subsets of a lattice, meaning any two elements in the interval have their join and meet also in the interval
• In the real numbers $\mathbb{R}$ with the usual order, intervals correspond to the familiar notions of intervals on the real line
• Examples of intervals in $\mathbb{R}$:
  • Closed interval: $[0, 1] = \{x \in \mathbb{R} : 0 \leq x \leq 1\}$
  • Open interval: $(2, 5) = \{x \in \mathbb{R} : 2 < x < 5\}$
  • Half-open intervals: $[1, 3) = \{x \in \mathbb{R} : 1 \leq x < 3\}$ and $(0, 2] = \{x \in \mathbb{R} : 0 < x \leq 2\}$

Interval Structures

Sublattices and Intervals

• A sublattice of an interval $I$ in a lattice $L$ is a subset $S \subseteq I$ that is itself a lattice under the induced order from $L$
• For $S$ to be a sublattice of $I$, it must be closed under the join and meet operations of $L$ restricted to $I$
• If $I = [a, b]$ is a closed interval in $L$, then $I$ is a sublattice of $L$ since it is closed under joins and meets
• Examples:
  • In $\mathbb{R}$, any closed interval $[a, b]$ is a sublattice of $\mathbb{R}$
  • In a Boolean algebra, any interval $[a, b]$ is a sublattice isomorphic to a Boolean algebra

Bounded Lattices and Lattice Segments

• A bounded lattice is a lattice $L$ with a least element $\bot$ (bottom) and a greatest element $\top$ (top)
• In a bounded lattice $L$, the entire lattice is the closed interval $[\bot, \top]$
• A lattice segment is a convex sublattice of a lattice $L$, i.e., an interval that is also a sublattice
• Examples:
  • In a finite chain (totally ordered set) with $n$ elements, any interval is a lattice segment
  • In the lattice of subsets of a set $X$ (the power set lattice), any interval $[A, B]$ where $A \subseteq B \subseteq X$ is a lattice segment isomorphic to the power set lattice of $B \setminus A$