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The Lumenary
بداية المرحلة الجامعيةFoundationsمُخطَّط له

Orders, lattices and well-orders

Partial and total orders, Hasse diagrams, maximal elements and suprema, lattices and well-orders, with Zorn's lemma stated as a tool to be justified later.

math.found.order-theory

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يعتمد على

Relations, equivalence relations and partitionsProof: direct, contrapositive, contradiction and induction

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 11

ستتعلّم

  • 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.

تعريف

  • 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.

مبرهنة

  • 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.

مفاهيم خاطئة شائعة

  • 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.

يقود إلى

Constructing ℤ, ℚ and ℝOrdinals and cardinals

اكتشافات اختيارية

  • 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.

مسارات التعلّم

قراءات إضافية

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