Set Theory/Orderings

Definitions
Relations with certain properties that impose a notion of order on a set are known as order relations or simply orderings. For the following definitions, let R be a binary relation.


 * If R is reflexive and transitive, then it is known as a preorder.
 * If R is a preorder and also antisymmetric, then it is known as a partial order.
 * If R is a partial order and also total, then it is known as a total order or a linear order.

A set equipped with a preorder, partial order, or total order is known as a preordered set, partially ordered set (or poset), or totally ordered set (or linearly ordered set) respectively. An order relation is usually denoted by the symbol $$\le$$ and an ordered set is denoted by the ordered pair $$( S, \le )$$ where $$\le$$ is the order relation on S.

A totally ordered subset of a partially ordered set is known as a chain. For this reason, any totally ordered set may sometimes be referred to as a chain.

Two elements a and b in a preordered (and thus in a partially or totally ordered) set are called comparable if either $$a \le b$$ or $$b \le a$$. Note that while totality guarantees that every two elements in a totally ordered set are comparable, two elements in a pre or partially ordered set may not be so.

Bounds
Let $$(S, \le)$$ be a preordered set and let $$T$$ be a subset of $$S$$. If there exists an element $$u$$ in $$S$$ such that $$x \le u$$ for all $$x \in T$$, then $$u$$ is called an upper bound for $$T$$. Similarly, if there exists an element $$l$$ in $$S$$ such that $$l \le x$$ for all $$x \in T$$ then $$l$$ is a lower bound for $$T$$. If there exists an upper bound for a set then that set is said to be bounded above, or similarly if there is a lower bound then the set is bounded below.

Let $$(P, \le)$$ be a partially ordered set and let $$T$$ be a subset of $$P$$. If an element $$u \in P$$ is an upper bound for $$T$$ and if $$u \le z$$ whenever $$z$$ is an upper bound for $$T$$ then $$u$$ is called the least upper bound or supremum of $$T$$. Similarly, a lower bound of $$T$$ that is greater than or equal to every other lower bound for $$T$$ is the greatest lower bound or infimum of $$T$$. The following proposition states that we are justified in calling these elements the supremum or infimum rather than just a supremum or infimum. The proof is left to the reader.

Proposition: The supremum and infimum of a set are each unique.

Let $$(P, \le)$$ be a partially ordered set and $$T$$ be a subset of $$P$$. A maximal element of $$T$$ is any element $$m$$ such that if $$m \le a$$ then $$m = a$$ for all $$a \in T$$. If the inequality in the previous sentence is reversed, then the element is called a minimal element. If $$t \in T$$ is greater than every other element in $$T$$, then $$t$$ is the greatest element or maximum, and similarly if it is less than every other element it is the least element or minimum. Note that an element of a partially ordered set can be a maximal element while failing to be a maximum since not all elements of a partially ordered set may be comparable.

Equivalences
Another important type of relation is the equivalence relation. This is a relation R that is reflexive, symmetric, and transitive (or, simply a preorder that is also symmetric). When R is an equivalence relation, we usually denote it by $$\sim$$ or $$\equiv$$. A set equipped with an equivalence relation is also known as a setoid.

If $$\sim$$ is an equivalence relation on a set $$S$$, we define for an element $$s \in S$$ the equivalence class of $$s$$ as $$\{a \in S : a \sim s\}$$. This is usually denoted by $$[s]$$. The set of all equivalence classes of $$S$$ is known as the quotient set of $$S$$ by $$\sim$$, which we denote by $$S/\!\sim\ = \{[x] : x \in S\}$$.

A partition of a set $$S$$ is a family of sets $$\mathfrak{S}$$ such that $$\mathfrak{S}$$ is pairwise disjoint and $$\bigcup \mathfrak{S} = S$$. The proof of the following theorems about equivalence relations are left to the reader.

Theorem: If $$S$$ is a set and $$\sim$$ is an equivalence relation on $$S$$, then $$S/\!\sim$$ is a partition of $$S$$.

Theorem: Let $$S$$ be a set and $$P$$ a partition of $$S$$. Define a relation $$\star$$ such that for $$a, b \in S$$, $$a \star b$$ holds if and only if there exists a member of P which contains both $$a$$ and $$b$$. Then, $$\star$$ is an equivalence relation.