Definition & Properties of Lattices
Introduction
A lattice is a mathematical structure that consists of a partially ordered set in which every pair of elements has a unique supremum (least upper bound) and a unique infimum (greatest lower bound).
Definition
A lattice L is defined as a partially ordered set (poset) in which every two elements have both a least upper bound (supremum) and a greatest lower bound (infimum), denoted by lub and glb respectively.
Properties of Lattices
1. Associativity
The join and meet operations in a lattice are associative, meaning that for any elements a, b, and c in the lattice, (a ∨ b) ∨ c = a ∨ (b ∨ c) and (a ∧ b) ∧ c = a ∧ (b ∧ c).
2. Commutativity
The join and meet operations are commutative, meaning that for any elements a and b in the lattice, a ∨ b = b ∨ a and a ∧ b = b ∧ a.
3. Absorption
The absorption laws state that for any elements a and b in the lattice, a ∨ (a ∧ b) = a and a ∧ (a ∨ b) = a.
4. Idempotent
Elements in a lattice are idempotent under both join and meet operations, meaning that a ∨ a = a and a ∧ a = a for any element a in the lattice.
5. Bounded
A lattice is bounded if it has a unique greatest element (top element) and a unique least element (bottom element). The top element is denoted by ⊤ (top) and the bottom element by ⊥ (bottom).
6. Distributivity
A lattice is distributive if it satisfies the distributive law, which states that for any elements a, b, and c in the lattice, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).