* properties of binary relation
let R be a binary relation in A.
R is
reflexive in A if aRa for all a ∈ A.
symmetric in A if aRb implies bRa for all a, b ∈ A.
antisymmetric in A if aRb and bRa imply a = b for all a, b ∈ A.
asymmetric in A if aRb implies that bRa does not hold for any a, b ∈ A.
transitive in A if aRb and bRc imply aRc for all a, b, c ∈ A.

* equivalence
R is an equivalence on A if it is reflexive, symmetric, and transitive.

* ordered set
let R be a relation in A.

- poset (partially ordered set)
R is a (partial) ordering of A if it is reflexive, antisymmetric, and transitive.
(A, R) is called an ordered set.

ex: "less than or equal to" relation
divisibility relation | : a|b iff a divides b.

- strict ordering
R is a strict ordering if it is asymmetric and transitive.
ex: "less than" relation

- loset (linearly ordered set)
an ordering of A is called linear or total if any two elements of A are comparable.
(A, R) is called a linearly ordered set.

( a and b are comparable in an (partial) ordering R if aRb or bRa; in a strict ordering if aRb or bRa or a = b)

ex: < is total, but | is not.

- woset (well ordered set)
linear ordering R of A is a well-ordering if every non-empty subset of A has a R-least element.
(A, R) is called a well-ordered set.

* notions of greatest and least
let ≤ be an (partial) ordering of A, and let B is a subset of A.

- least/greatest/minimal/maximal
b ∈ B is the least element of B in the ordering ≤ if b≤x for every x ∈ B.
b ∈ B is the greatest element of B in ≤ if x≤b for every x ∈ B

b ∈ B is a minimal element of B in ≤ if there exists no x ∈ B such that x≤b and x != b.
b ∈ B is a maximal element of B in ≤ if there is no x ∈ B such that b≤x and x != b.

ex: B = positive integers greater than 1 = {2, 3, 4, ...} does not have a least element in |, but it has many minimal elements (for example, 2, 3, 5, etc)

- lower bound/upper bound/infimum/supremum
a ∈ A is a lower bound of B if a≤x for all x ∈ B.
a ∈ A is an upper bound of B if x≤a for all x ∈ B.

a ∈ A is an infimum (or greatest lower bound) of B if it is the greatest element of the set of all lower bounds of B.
a ∈ A is a supremum (or least upper bound) of B if it is the least element of the set of all upper bounds of B.