5.1 Complete lattices and their properties

Complete lattices are powerful structures in lattice theory. They ensure every subset has a least upper bound and greatest lower bound, providing a solid foundation for analyzing ordered sets and their properties.

Lattice homomorphisms and isomorphisms help us understand relationships between different lattices. We'll explore these concepts, along with suprema, infima, chains, and directed sets, to grasp the full scope of complete lattices.

Complete Lattices

Definition and Properties of Complete Lattices

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• Complete lattice has every subset containing a least upper bound (supremum) and greatest lower bound (infimum)
• Completeness property ensures the existence of suprema and infima for all subsets of the lattice
• Completeness axiom states that a lattice $L$ is complete if every subset $S \subseteq L$ has a supremum and an infimum in $L$
  • For example, the power set of a set ordered by inclusion $(P(X), \subseteq)$ forms a complete lattice
  • The set of real numbers $\mathbb{R}$ with the usual order $\leq$ is not a complete lattice, as the subset of positive real numbers has no greatest element

Lattice Homomorphisms

• Lattice homomorphism is a function $f: L_1 \rightarrow L_2$ between two lattices that preserves the lattice operations (join and meet)
  • $f(a \vee b) = f(a) \vee f(b)$ and $f(a \wedge b) = f(a) \wedge f(b)$ for all $a, b \in L_1$
• Lattice homomorphisms preserve the order relation between elements
  • If $a \leq b$ in $L_1$, then $f(a) \leq f(b)$ in $L_2$
• Lattice isomorphism is a bijective lattice homomorphism, indicating that two lattices have the same structure
  • For instance, the lattice of divisors of 12 and the lattice of divisors of 18 are isomorphic

Bounds and Sets

Supremum and Infimum

• Supremum (least upper bound) of a subset $S$ of a partially ordered set $(P, \leq)$ is the least element in $P$ that is greater than or equal to all elements of $S$
  • Denoted as $\sup S$ or $\bigvee S$
  • For example, in the lattice of real numbers with the usual order, $\sup\{1, 2, 3\} = 3$
• Infimum (greatest lower bound) of a subset $S$ of a partially ordered set $(P, \leq)$ is the greatest element in $P$ that is less than or equal to all elements of $S$
  • Denoted as $\inf S$ or $\bigwedge S$
  • For example, in the lattice of real numbers with the usual order, $\inf\{1, 2, 3\} = 1$

Chains and Directed Sets

• Chain is a totally ordered subset of a partially ordered set
  • For any two elements $a, b$ in a chain, either $a \leq b$ or $b \leq a$
  • An example of a chain in the lattice of subsets of $\{1, 2, 3\}$ is $\{\emptyset, \{1\}, \{1, 2\}\}$
• Directed set is a partially ordered set $(D, \leq)$ in which any two elements have an upper bound in $D$
  • For any $a, b \in D$, there exists $c \in D$ such that $a \leq c$ and $b \leq c$
  • The set of natural numbers $\mathbb{N}$ with the usual order is a directed set, as for any $a, b \in \mathbb{N}$, $\max\{a, b\}$ is an upper bound