Orders, lattices and well-orders

Se basa en

Requisitos previos obligatorios, hasta el fondo: 11

Aprenderás a

• Verify the partial-order axioms and draw Hasse diagrams for finite posets such as divisibility on {1, …, 12} and inclusion on P({1, 2, 3}).
• Distinguish maximal from greatest elements, and upper bounds from least upper bounds, with examples.
• Define lattices and well-orders, and show that ℕ is well-ordered by ≤ while ℤ and ℚ are not.

Definición

• A partial order on P is a relation ≤ that is reflexive, antisymmetric and transitive.
• A lattice is a poset in which every pair of elements has a least upper bound (join) and a greatest lower bound (meet).
• A well-order is a total order in which every non-empty subset has a least element.
• Zorn's lemma: if every chain in a non-empty poset has an upper bound, then the poset has a maximal element.

Teorema

• Every finite non-empty poset has a maximal element.
• (P(A), ⊆) is a lattice with join ∪ and meet ∩.
• Over ZF, Zorn's lemma is equivalent to the axiom of choice.

Errores frecuentes

• Treating 'maximal' and 'greatest' as synonyms.
• Assuming every partial order is total.
• Thinking a least upper bound of a subset must belong to that subset.

Conduce a

Descubrimientos opcionales

• Categories A preorder is exactly a category with at most one morphism between any two objects; order-preserving maps between preorders are then exactly functors.
• Algebraic structures: monoids, groups and rings A lattice can equivalently be described as a set with two operations ∧ and ∨ satisfying associative, commutative and absorption laws.

Itinerarios de aprendizaje

• Undergraduate mathematics major

Lecturas complementarias

• B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (1990)