Partial Order Relations On A Lattice

Partial Order Relations on a Lattice
A partial order relation on a lattice (L) follows as a consequence of the axioms for the binary operations ∨ and ∧ .
We define a relation ≤ on L such that for a, b ∈ L ,
a ≤ b <-> a ∨ b = b
or analogously,
a ≤ b <-> a ∧ b = a .
We note that
(i) For any a ∈ L
a ∨ a = a (idempotent law), therefore a ≤ a showing that ≤ is reflexive.
(ii) Let a ≤ b and b ≤ a. Therefore
a ∨ b = b
b ∨ a = a
But
a ∨ b = b ∨ a (Commutative Law in lattice)
Hence
a = b ,
showing that ≤ is antisymmetric.
(iii) Suppose that a ≤ b and b ≤ c. Therefore a ∨ b = b and b ∨ c = c . Then
a ∨ c = a ∨ (b ∨ c)
= (a ∨ b) ∨ c (Associativity in lattice)

= b ∨ c
= c ,
showing that a ≤ c and hence ≤ is transitive.
This shows that a lattice is a partially ordered set