4.4 Dense and discrete lattices

Dense and discrete lattices are two extremes in lattice theory. Dense lattices have no gaps between elements, like the real numbers. Discrete lattices have no elements between consecutive ones, like integers.

These concepts connect to special elements in lattices. Dense lattices have infinite chains between elements, while discrete lattices have minimal distances between comparable elements. Understanding these helps us grasp lattice structure and properties.

Density and Discreteness

Dense Lattices

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• A lattice $(L, \leq)$ is dense if for any two elements $a, b \in L$ with $a < b$, there exists an element $c \in L$ such that $a < c < b$
• Intuitively, a dense lattice has no "gaps" between its elements
• In a dense lattice, between any two distinct comparable elements, there is always another element
• The real numbers $(\mathbb{R}, \leq)$ form a dense lattice
  • For any two real numbers $a < b$, we can always find a real number $c$ such that $a < c < b$ (e.g., $c = \frac{a+b}{2}$)

Discrete Lattices

• A lattice $(L, \leq)$ is discrete if for any two elements $a, b \in L$ with $a < b$, there is no element $c \in L$ such that $a < c < b$
• In a discrete lattice, there are no elements between any two distinct comparable elements
• The integers $(\mathbb{Z}, \leq)$ form a discrete lattice
  • For any two integers $a < b$, there is no integer $c$ such that $a < c < b$
• The power set of a set ordered by inclusion $(P(S), \subseteq)$ is a discrete lattice
  • For any two distinct subsets $A, B \in P(S)$ with $A \subset B$, there is no subset $C \in P(S)$ such that $A \subset C \subset B$

Density Property

• The density property states that a lattice $(L, \leq)$ is dense if and only if for any two elements $a, b \in L$ with $a < b$, there exists an infinite chain $C \subseteq L$ such that $a = \inf C$ and $b = \sup C$
• This property characterizes dense lattices using the existence of infinite chains between any two distinct comparable elements
• The density property provides an alternative definition of dense lattices using infima and suprema of chains

Order Structures

Chains

• A chain in a partially ordered set $(P, \leq)$ is a subset $C \subseteq P$ such that any two elements in $C$ are comparable
• In a chain, every pair of elements is related by the order relation
• A chain is a totally ordered subset of a partially ordered set
• Examples of chains include $(\mathbb{N}, \leq)$, $(\mathbb{Z}, \leq)$, and $(\mathbb{Q}, \leq)$

Antichains

• An antichain in a partially ordered set $(P, \leq)$ is a subset $A \subseteq P$ such that no two elements in $A$ are comparable
• In an antichain, no pair of elements is related by the order relation
• An antichain is a set of mutually incomparable elements
• Examples of antichains include the set of minimal elements in a lattice and the set of maximal elements in a lattice

Order Topology

• The order topology on a lattice $(L, \leq)$ is the topology generated by the subbase consisting of sets of the form $\{x \in L : x < a\}$ and $\{x \in L : x > a\}$ for each $a \in L$
• The order topology is a natural way to define a topology on a lattice using the order relation
• Open sets in the order topology are unions of intervals of the form $(a, b) = \{x \in L : a < x < b\}$
• The order topology is always Hausdorff and has a basis consisting of intervals

Lattice Properties

Completeness

• A lattice $(L, \leq)$ is complete if every subset $S \subseteq L$ has a least upper bound (supremum) and a greatest lower bound (infimum) in $L$
• Completeness is a strong property that ensures the existence of joins and meets for arbitrary subsets, not just finite ones
• In a complete lattice, every subset has a supremum and an infimum, which are elements of the lattice
• Examples of complete lattices include:
  • The power set of a set ordered by inclusion $(P(S), \subseteq)$
  • The set of all closed intervals of real numbers ordered by inclusion $(\{[a, b] : a, b \in \mathbb{R}, a \leq b\}, \subseteq)$
• Complete lattices have many desirable properties and are important in various areas of mathematics, such as topology and order theory