本科低年级Foundations计划中
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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前置知识
Relations, equivalence relations and partitionsProof: direct, contrapositive, contradiction and induction
11 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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.
定理
- 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.
常见误解
- 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.
通向
可选发现
- 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.
学习路径
延伸阅读
- B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (1990)