## Partially Ordered SetsConsider a relation R on a set S satisfying the following properties: - R is reflexive, i.e., xRx for every x ∈ S.
- R is antisymmetric, i.e., if xRy and yRx, then x = y.
- R is transitive, i.e., xRy and yRz, then xRz.
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:- The set N of natural numbers form a poset under the relation '≤' because firstly x ≤ x, secondly, if x ≤ y and y ≤ x, then we have x = y and lastly if x ≤ y and y ≤ z, it implies x ≤ z for all x, y, z ∈ N.
- The set N of natural numbers under divisibility i.e., 'x divides y' forms a poset because x/x for every x ∈ N. Also if x/y and y/x, we have x = y. Again if x/y, y/z we have x/z, for every x, y, z ∈ N.
- Consider a set S = {1, 2} and power set of S is P(S). The relation of set inclusion ⊆ is a partial order. Since, for any sets A, B, C in P (S), firstly we have A ⊆ A, secondly, if A ⊆B and B⊆A, then we have A = B. Lastly, if A ⊆B and B ⊆C,then A⊆C. Hence, (P(S), ⊆) is a poset.
## Elements of POSET:**Maximal Element:**An element a ∈ A is called a maximal element of A if there is no element in c in A such that a ≤ c.**Minimal Element:**An element b ∈ A is called a minimal element of A if there is no element in c in A such that c ≤ b.
## Note: There can be more than one maximal or more than one minimal element.
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 ## Non-Comparable Elements:Consider an ordered set A. Two elements a and b of set A are called non-comparable if neither a ≤ b nor b ≤ a.
The non-comparable 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.
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