Describe Partially Ordered Set

A partially ordered set (or poset) is a set taken together with a partial order. Formally, a partially ordered set is defined as an ordered pair P =(X,≤) where X is called the ground set of P and ≤ is the partial order of P.

An element u in a partially ordered set (X,≤) is said to be an upper bound for a subset S of X if, for every s ∈ S, we have s ≤ u

Consider a relation R on a set S satisfying the following properties:

  1. R is reflexive, i.e., xRx for every x ∈ S.
  2. R is antisymmetric, i.e., if xRy and yRx, then x = y.
  3. R is transitive, i.e., xRy and yRz, then xRz.

Then R is called a partial order relation, and the set S together with a partial order is called a partial order set or POSET and is denoted by (S ≤).

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