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Home / Lattices and Boolean Algebras / ISOMORPHISM | LATICEES AND BOOLEAN ALGEBRAS

ISOMORPHISM | LATICEES AND BOOLEAN ALGEBRAS

Isomorphism: Let (P, Q) and (Q, R′) be two posets. A one-one onto map

f : P→Q is called an isomorphism if x R y ⇔f (x) R′ f (x), x, y ∊P.

Isotone: A map f : P→Q is called an isotone or order preserving if x≤y ⟹ f (x) ≤ f (y)

Theorem: A mapping f : P→Q is an isomorphism iff is isotone and has an inverse.

Proof: Let f : P→Q be an isomorphism.

Thus being 1-1, onto, f-1exists and is 1-1, onto.

Again by definition of isomorphism, f will be isotone. We show                   f -1 : Q→P is also isotone.

Let y1, y2 ∊ Q where y1 ≤ y2. Since f is onto, there exist x1, x2 ∊ P, such that f (x1) = y1, f (x2) = y2 ⇔ x1 = f 1(y1), f 1(y2)

Now y1 ≤ y2f (x1) ≤ f (x2)

⟹ x1 ≤ x2

f -1(y1) ≤ f -1

f -1 is isotone.

 

Conversely, let f be isotone such that f -1 is also isotone. Since f -1 exists, f is one-one, onto.

Again, as f is isotone x1 ≤ x2 ⟹ f(x1) ≤ f(x2), x1, x2 ∊ P

Also f -1 is isotone implies

f (x1) ≤ f (x2) ⟹ f -1(f (x1)) ≤ f -1( f(x1))

⟹ x1 ≤ x2

Thus            x1 ≤ x2f (x1) ≤ f (x2)

Hence f is an isomorphism.        (Proved)

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