Poset: A non empty set P, together with a binary relation R is said to form a partially ordered set or a poset if the following conditions hold:
P1: Reflexivity: aRa for all a∊ P
P2: Anti – symmetry: If aRb, bRa then a = b ( a, b ∊P)
P3: Transitivity: If aRb, bRc then aRc (a, b, c ∊P)
Example: The set N of natural numbers forms a poset under the usual ≤. Similarly, the integers, rationals and real numbers also form posets under usal ≤.
Comparable: If a≤b in a poset, we say a and b are comparable. Two elements of a poset may or may not be comparable. If a ≤ b and a ≠ b, we will write a<b.
Chain: If P is a poset in which every two members are comparable it is called a totally ordered set or a toset or a chain.