Order Theory
Order theory the concept of order when using binary relations
Pre Orders
A relation \(R\) on \(A\) is called a preorder of \(A\) if its reflexive and transitive.
Partial Orders
A partial order describes the order of some of the ordered pairs in a relation. \(R\) is a partial order on \(A\) if its reflexive, transitive, and antisymmetric. A partial order is also a preorder.
Notice that \(\exists x \in A \exists y \in A ((x, y) \notin R)\), and that’s the reason why a partial order doesn’t provide a way to determine the order of any elements of \(A\).
Total Orders
A total order describes the order of all of the ordered pairs in a relation. \(R\) is a total order on \(A\) if its already a partial order, and if \(\forall x \in A \forall y \in A (xRy \lor yRx)\).
Strict Partial Orders
A strict partial order describes the order of some of the ordered pairs in a relation, but making sure that pairs consisting of the same element, and mirrored pairs are omitted. \(R\) is a strict partial order on \(A\) if its irreflexive, transitive, and asymmetric (which assumes the relation is antisymmetric as well). Keep in mind a strict partial order is not a partial order.
Strict Total Orders
A strict total order extends a strict partial order to describe the order of all of the ordered pairs in a relation. \(R\) is a strict total order on \(A\) if its already a strict partial order, and if \( \forall x \in A \forall y \in A (xRy \lor yRx \lor x = y) \).
Special Elements
Given a binary relation \(R \subseteq A \times A\) an arbitrary element \(x \in A\):

Minimal: \(x\) is a minimal element of \(R\) if \(\lnot \exists a \in A ((a, x) \in R \land a \neq x)\)

Maximal: \(x\) is a maximal element of \(R\) if \(\lnot \exists a \in A ((x, a) \in R \land a \neq x)\)

Smallest: \(x\) is the smallest element of \(R\) if \(\forall a \in A ((x, a) \in R)\)
Note that the smallest element is also a minimal element. Since a total order describes the ordering of all possible elements, then if \(R\) is a total order and \(x\) is a minimal element, then its also the smallest one.
 Largest: \(x\) is the largest element of \(R\) if \(\forall a \in A ((a, x) \in R)\)
Note that the largest element is also a maximal element. Since a total order describes the ordering of all possible elements, then if \(R\) is a total order and \(x\) is a maximal element, then its also the largest one.
Given a partian order \(R\) on \(A\), \(B \subseteq A\), and \(x \in A\):
 Lower bound: \(x\) is a lower bound for \(B\) if \(\forall b \in B ((x, b) \in R)\)
Note that the smallest element of \(B\) is a lower bound that is also an element of \(B\).

Upper bound: \(x\) is a lower bound for \(B\) if \(\forall b \in B ((b, x) \in R)\)
 Least upper bound: The smallest element of the upper bounds
 Greatest lower bound: The largest element of the lower bounds