Lattice Isomorphism
The lattice so obtained is named Bn. The properties of the partial order in Bn can be described directly as follows:
Let x = a1 a2…..an and y = b1 b2…..bn be any two elements of Bn. Then
(1) x ≤ y if and only if ak < bk, k = 1, 2,…..,n, where ak and bk are 0 or 1.
(2) x ∧ y = c1 c2….cn, where ck = min(ak, bk).
(3) x ∨ y = d1 d2 ….dn, where dk = max(ak, hk).
(4) x has a complement x' = z1 z2……zn where zk = 1 if xk = 0 and zk = 0 if xk = 1.
Remark: (Bn, ≤) under the partial order £ defined above is isomorphic to
(P(A), ⊆), when A has n elements. In such a case x ≤ y corresponds to S ⊆ T, x ∨ y corresponds to S È T and x' corresponds to Ac.
Example : Let D6 = {1, 2, 3, 6}, set of divisors of 6. Then D6 is isomorphic to B2. In fact f : D6 -> B2 defined by
f(1) = 00, f(2) = 10, f(3) = 01, f(6) = 11
is an isomorphism.
Example: Let A = {a, b} and P(A) = {φ, {a}, {a, b}} then the lattice (P(A), ⊆) is isomorphic to the lattice (D6, 1) with divisibility as the partial order relation. In fact, we define a mapping f : D6 -> P(A) by
f(1) = φ, f(2) = {a}, f(3) = {b}, f(6) = {a, b} ,
then f is bijective and we note that
1|2 <-> {j} Í {a} <-> f(1) ⊆ f(2)
2|6 <-> {a} Í {a, b} <-> | f(2) ⊆ f(6) and so on.
Hence f is isomorphism.
Definition: Let (L, ∨ , ∧ ) be a lattice. Then lattice homomorphism f : L -> L is called an endomorphism.
Definition: Let (L, ∨ , ∧ ) be lattice. Then the lattice isomorphism f: L->L is called an automorphism.
If f : L -> L is an endomorphism, then the image set of f is sublattice of L.
Definition: Let (A, ≤) and (B, ≤') be two partially ordered sets. A mapping f :
A -> B is called order preserving relative to the ordering ≤ in A and ≤' in B iff for a, b ∈ A, a ≤ b f (a) ≤' f (b)
If A and B are lattices and f : A -> B is a lattice homomorphism, then f is order preserving.
Definition: Two partially ordered sets (A, ≤) and (B, ≤,) are said to be order isomorphic if there exists a mapping f : A -> B which is bijective and both f and f-1 are order preserving.
For lattices (A, ≤) and (B, ≤'), an order isomorphism is equivalent to lattice isomorphism. Hence lattices which are order-isomorphic as partially ordered sets are isomorphic.
Let (L, ∨ , ∧ ) be a lattice and let S = {a1, a2….,an} be a finite subset of L. Then
LUB of S is represented by a1 ∨ a2 ∨…..∨ an
GLB of S is represented by a1 ∧ a2 ∧……∧ an
Definition: A lattice is called complete if each of its non-empty subsets has a least upper bound and a greatest lower bound.
Obviously, every finite lattice is complete.
Also every complete lattice must have a least element, denoted by 0 and a greatest element, denoted by I. The least and greatest elements if exist are called bound (units, universal bounds) of the lattice.