11.3 Lattices in universal algebra

Lattices play a crucial role in universal algebra, connecting abstract structures to concrete mathematical concepts. This section explores varieties, equational classes, and congruence relations, showing how lattices help organize and analyze algebraic systems.

Free algebras, subdirect products, and Birkhoff's theorem are key tools for understanding algebraic structures. These concepts demonstrate how lattices provide a framework for studying relationships between different algebraic systems and their properties.

Varieties and Equational Classes

Defining Varieties and Equational Classes

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• Variety is a class of algebras that is closed under homomorphic images, subalgebras, and direct products
  • Can be defined by a set of identities (equations) that hold in all algebras in the class
  • Examples include the variety of groups, rings, and lattices
• Equational class is another term for a variety
  • Emphasizes that the class can be defined by a set of equations (identities)
  • Every variety is an equational class and vice versa

Special Types of Varieties

• Congruence distributive variety is a variety in which the lattice of congruences of each algebra is distributive
  • Distributivity means that for any congruences $\theta$, $\phi$, and $\psi$, we have $\theta \wedge (\phi \vee \psi) = (\theta \wedge \phi) \vee (\theta \wedge \psi)$
  • Examples include the variety of lattices and the variety of Boolean algebras
• Congruence modular variety is a variety in which the lattice of congruences of each algebra is modular
  • Modularity is a weaker condition than distributivity
  • For any congruences $\theta$, $\phi$, and $\psi$ with $\theta \leq \psi$, we have $(\theta \vee \phi) \wedge \psi = \theta \vee (\phi \wedge \psi)$
  • Examples include the variety of groups and the variety of rings

Congruence Relations

Congruence Lattice

• Congruence lattice of an algebra $A$ is the lattice of all congruence relations on $A$
  • Congruence relations are equivalence relations that are compatible with the operations of the algebra
  • The congruence lattice is a complete lattice under inclusion
  • Plays a crucial role in understanding the structure of an algebra

Maltsev Conditions

• Maltsev condition is a property of a variety that can be expressed in terms of the existence of certain term operations
  • Used to characterize various properties of varieties, such as congruence distributivity or modularity
  • Example: A variety is congruence permutable if and only if there exists a term $p(x, y, z)$ such that $p(x, y, y) = x$ and $p(x, x, y) = y$ hold in the variety
  • Congruence permutability means that the composition of any two congruences is equal to their join

Algebraic Constructions

Free Algebras

• Free algebra in a variety $\mathcal{V}$ over a set $X$ is an algebra $F_\mathcal{V}(X)$ in $\mathcal{V}$ with a map $i: X \to F_\mathcal{V}(X)$ such that for any algebra $A$ in $\mathcal{V}$ and any map $f: X \to A$, there exists a unique homomorphism $\bar{f}: F_\mathcal{V}(X) \to A$ with $\bar{f} \circ i = f$
  • $X$ is called the set of generators, and $i$ is the inclusion map
  • Free algebras have the universal mapping property
  • Examples include free groups, free lattices, and free Boolean algebras

Subdirect Products

• Subdirect product of a family of algebras $(A_i)_{i \in I}$ is a subalgebra of the direct product $\prod_{i \in I} A_i$ that projects onto each factor $A_i$
  • Generalizes the notion of a subdirect sum
  • Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras
  • Subdirectly irreducible algebras cannot be decomposed as a subdirect product of non-trivial algebras

Birkhoff's Theorem

• Birkhoff's theorem states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and direct products
  • Provides a fundamental characterization of varieties
  • Allows for the construction of new varieties from existing ones
  • Example: The class of all lattices is a variety because it is closed under homomorphic images, subalgebras, and direct products