Partially Ordered SetsConsider a relation R on a set S satisfying the following properties:
Then R is called a partial order relation, and the set S together with partial order is called a partially order set or POSET and is denoted by (S, ≤). Example:
Elements of POSET:
Note: There can be more than one maximal or more than one minimal element.Example: Determine all the maximal and minimal elements of the poset whose Hasse diagram is shown in fig: Solution: The maximal elements are b and f. The minimal elements are d and e. Comparable Elements:Consider an ordered set A. Two elements a and b of set A are called comparable if a ≤ b or b ≤ a NonComparable Elements:Consider an ordered set A. Two elements a and b of set A are called noncomparable if neither a ≤ b nor b ≤ a. Example: Consider A = {1, 2, 3, 5, 6, 10, 15, 30} is ordered by divisibility. Determine all the comparable and noncomparable pairs of elements of A. Solution: The comparable pairs of elements of A are: The noncomparable pair of elements of A are: Linearly Ordered Set:Consider an ordered set A. The set A is called linearly ordered set or totally ordered set, if every pair of elements in A is comparable. Example: The set of positive integers I_{+} with the usual order ≤ is a linearly ordered set.
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