🔳Lattice Theory Unit 9 – Free Lattices and Whitman's Condition

Key Concepts and Definitions

• Lattice: An algebraic structure consisting of a partially ordered set in which every pair of elements has a unique least upper bound (join) and greatest lower bound (meet)
• Free Lattice: A lattice generated by a set of elements without any additional relations beyond those required by the lattice axioms
  • Generated by a set of elements called free generators
  • Satisfies the universal property for lattices
• Join: The least upper bound of two elements in a lattice, denoted by the symbol $\vee$
• Meet: The greatest lower bound of two elements in a lattice, denoted by the symbol $\wedge$
• Partially Ordered Set (Poset): A set equipped with a binary relation that is reflexive, antisymmetric, and transitive
• Whitman's Condition: A property of lattices that characterizes free lattices, stating that if $a \wedge b \leq c \vee d$, then either $a \leq c \vee d$, $b \leq c \vee d$, $a \wedge b \leq c$, or $a \wedge b \leq d$
• Universal Property: A property that uniquely characterizes an object or structure up to isomorphism

Free Lattices: Introduction and Properties

• Free lattices generated by a set of elements without imposing any additional relations beyond the lattice axioms
• Satisfy the universal property for lattices, which states that for any lattice $L$ and any function $f$ from the set of free generators to $L$, there exists a unique lattice homomorphism extending $f$ to the entire free lattice
• Can be constructed using the Funayama-Nakayama construction, which involves forming equivalence classes of terms built from the free generators using the join and meet operations
• Enjoy a unique factorization property, where each element can be uniquely represented as a join of meet-irreducible elements or a meet of join-irreducible elements
• Play a crucial role in the study of lattice varieties and equational theories of lattices
• Serve as a tool for understanding the structure and properties of general lattices
• Whitman's condition provides a characterization of free lattices among the class of all lattices

Construction of Free Lattices

• Funayama-Nakayama construction: A method for constructing the free lattice generated by a set $X$
  1. Consider the set $T(X)$ of all terms built from elements of $X$ using the join ($\vee$) and meet ($\wedge$) operations
  2. Define an equivalence relation $\equiv$ on $T(X)$ by $t_1 \equiv t_2$ if and only if $t_1$ and $t_2$ are provably equal using the lattice axioms
  3. The free lattice $FL(X)$ is the set of equivalence classes of terms under $\equiv$, with join and meet operations induced by the corresponding operations on terms
• The elements of the free lattice can be represented by their canonical forms, which are the shortest terms in each equivalence class
• The free lattice generated by a set $X$ satisfies the universal property: For any lattice $L$ and any function $f: X \to L$, there exists a unique lattice homomorphism $\hat{f}: FL(X) \to L$ extending $f$
• Free lattices are uniquely determined up to isomorphism by their sets of free generators
• The construction of free lattices allows for the study of lattice identities and equational theories

Whitman's Condition: Overview

• A property of lattices that characterizes free lattices among the class of all lattices
• States that for any elements $a$, $b$, $c$, and $d$ in a lattice $L$, if $a \wedge b \leq c \vee d$, then at least one of the following holds:
  • $a \leq c \vee d$
  • $b \leq c \vee d$
  • $a \wedge b \leq c$
  • $a \wedge b \leq d$
• Equivalent to the statement that every sublattice of $L$ generated by two elements is distributive
• A lattice satisfies Whitman's condition if and only if it is isomorphic to a sublattice of a free lattice
• Provides a useful tool for studying the structure and properties of free lattices
• Can be used to prove that certain lattices are not free by showing that they violate Whitman's condition

Applying Whitman's Condition

• To check if a lattice satisfies Whitman's condition, consider all possible combinations of four elements $a$, $b$, $c$, and $d$, and verify that the condition holds for each combination
• If a lattice violates Whitman's condition, it cannot be a free lattice or a sublattice of a free lattice
  • Example: The diamond lattice $M_3$ (consisting of four elements: bottom, top, and two incomparable elements) violates Whitman's condition and is not a sublattice of any free lattice
• Whitman's condition can be used to prove that certain lattice identities hold in all free lattices
  • Example: The modular law $(x \wedge y) \vee (y \wedge z) = y \wedge ((x \wedge y) \vee z)$ holds in all free lattices, as it can be derived using Whitman's condition
• Checking Whitman's condition can help determine if a given lattice has a free lattice representation or if it can be embedded into a free lattice
• Whitman's condition provides a practical tool for studying the structure and properties of lattices, particularly in relation to free lattices

Relationships Between Free Lattices and Whitman's Condition

• Whitman's condition characterizes free lattices: A lattice is free if and only if it satisfies Whitman's condition
• Every sublattice of a free lattice also satisfies Whitman's condition
  • Consequently, if a lattice violates Whitman's condition, it cannot be embedded into a free lattice
• The class of lattices satisfying Whitman's condition is closed under sublattices, products, and directed colimits
• Free lattices generate the variety of all lattices satisfying Whitman's condition
  • This variety is the smallest non-trivial lattice variety and is denoted by $\mathcal{FL}$
• The study of free lattices and Whitman's condition is closely related to the study of lattice varieties and equational theories of lattices
  • Whitman's condition provides a tool for understanding the structure and properties of lattice varieties
• The connection between free lattices and Whitman's condition allows for the application of universal algebraic techniques to the study of lattices

Practical Applications and Examples

• Free lattices used in the study of formal concept analysis, where they represent the lattice of formal concepts derived from a formal context
  • Example: In a formal context consisting of objects and attributes, the free lattice generated by the attributes represents all possible combinations of attributes and their implications
• Whitman's condition can be applied to analyze the structure of concept lattices and determine if they have a free lattice representation
• Free lattices appear in the study of lattice-valued logics, where they serve as the algebraic semantics for certain logical systems
  • Example: The Lindenbaum-Tarski algebra of a propositional logic is a free lattice generated by the set of propositional variables
• Whitman's condition can be used to characterize the lattice of subgroups of a free group
  • The subgroup lattice of a free group satisfies Whitman's condition and is isomorphic to a sublattice of a free lattice
• Free lattices and Whitman's condition find applications in the study of lattice-valued fuzzy logic and fuzzy set theory
  • The lattice of fuzzy sets forms a free lattice, and Whitman's condition can be used to study its properties

Advanced Topics and Further Study

• The study of free lattices and Whitman's condition leads to various advanced topics in lattice theory and universal algebra
• Lattice varieties and equational theories:
  • Free lattices generate the variety of all lattices satisfying Whitman's condition
  • The study of lattice varieties and their equational theories is closely related to the study of free lattices and Whitman's condition
• Connections with other algebraic structures:
  • Free lattices and Whitman's condition have connections with other algebraic structures, such as free monoids, free groups, and free rings
  • The study of these connections leads to a deeper understanding of the role of free objects in universal algebra
• Generalizations of Whitman's condition:
  • There are various generalizations of Whitman's condition, such as the meet-semidistributive law and the join-semidistributive law
  • These generalizations lead to the study of different classes of lattices and their relationships with free lattices
• Computational aspects:
  • The construction of free lattices and the verification of Whitman's condition have computational aspects, such as the efficient generation of canonical forms and the complexity of deciding lattice identities
  • The study of these computational aspects is important for the practical application of free lattices and Whitman's condition