Notes on Π 10 classes for Math 661 Fall 2002 Notre Dame University

1 Definitions and basic examples of Boolean algebras Definition 1.1. A partially ordered set (or poset for short) is a pair X, ≤ X where X is a nonempty set, ≤ X is a binary relation on X with the following properties for all x, y, z ∈ X: If, in addition, X, ≤ X satisfies the property (x ≤ X y ∨ y ≤ X x) for all x and y, then it is called a total or linear order. We frequently drop the subscript X from ≤ X as long as the underlying set is clear from the context. If a partial order has a least element, then we denote this element by 0 and if it has a greatest element, then we denote this element by 1. To be more formal, these elements are defined by the following properties: ∀x ∈ X(0 ≤ x) and ∀x ∈ X(x ≤ 1). Be careful not to confuse least elements with minimal elements or greatest elements with maximal elements. An element u ∈ X is minimal if it is not the case that ∃x ∈ X(x < u) and an element v ∈ X is maximal if it is not the case that ∃x ∈ X(v < x). Definition 1.2. Let X be a poset and A ⊂ X. The element x ∈ X is an upper bound for A if ∀a ∈ A(a ≤ x) and x is a lower bound for A if ∀a ∈ A(x ≤ a). A chain in a poset X is a linear ordered subset of X. Recall that Zorn's Lemma states that if every chain in a nonempty poset X has an upper bound, then X has a maximal element. These notes have been compiled from many sources, none of which are adequately cited. I claim no authorship of the results contained within and I am happy to supply detailed references for the sources of the material contained within these notes. For a breif description of some of the sources, see Section 16.