Project July 2002

Q.4. (a). Prove that the following version of the modular inequality :

aÚ[bÙ(a Ú b) < (a Ú b) Ù (a Ú c)]
for all a, b, c, ÎL, a lattice

Ans.

a Ú [b Ù (a Ú b) < (a Ú b) Ù (a Ú c)]
a Ú (b Ù c) < b Ù c
a Ú b < b
b < b Proved.
Note : a Ú b = b
a Ù b = a

Q. 4.(b). Define the following concepts from the field of lattice theory, with at least one example for each:

(i) Complete lattice
(ii) Distributive lattice
(iii) Modular lattice

Ans.

(i). Complete Lattice

Definition :- A lattice L is said to be complete if every non-empty subset has a least upper bound and greatest lower bound.

In view of our observations above we can say that:

(i) Every finite lattice is complete (Because every subset here is finite).

(ii) (IN, =), (Z, =), (R, =) etc. are not complete.

(iii) (IN, divides) is not complete.

Example :- Let S be any set (finite or infinite) and let L = (P(s), È, Ç). Then given any f ¹ US' |S'

A Í L, we have S' A and S' A are respectively, the upper bounds and greatest lower bound of A. these are simply the sub-sets of S consisting of all elements of S belonging to at least one element of a and that of all the elements of S belonging to each element of a respectively

Thus L is a complete lattice.

(ii) Distributive lattice

Theorem : In any lattice (L, Ú, Ù) the following statements are equivalent:

(i) a Ù (b Ú c) = (a Ù b) Ú (a Ù b) " a, b, c Î L

(ii) a Ú (a Ù b) = (a Ú b) Ù (a Ú c) " a, b, c Î L

Proof :- (i) => (ii)

R.H.S. of (ii) = [(a Ú b) Ù a] Ú [(a Ú b) Ù c] by (i)

= a Ú [c Ù (a Ú b)] by commutativity and absorption

= a Ú [(c Ù a) Ú (c Ù b)] by (i)

= [a Ú (c Ù a)] Ú (c Ù b) by associativity

= a Ú (b Ù c) by absorption and commutativity

= L.H.S. of (ii)

(ii) => (i) may be proved in a dual manner.

Definition :- A lattice (L, Ù, Ú ) is said to be distributive if the equivalent conditions of the above theorem hold.

Example :- L = (P(S), Ç, È) is distributive lattice, since the above distributive laws for Ç over È & È over Ç are well known facts of set theo0ry which are themselves consequences of distributivity of conjunction and disjunction of statements over each other in statement calculus.

(iii). Modular Lattice

Definition :- A lattice (L, =) is said to be modular if a Ú (b Ù c) = (a Ú b) Ù (a Ú c) = (a Ú b) Ù c as required.

But there are modular lattice which are not distributive.

Example :- The diamond lattice is a non-distributive modular lattice. We have already seen that it is not distributive.

Now the modularity condition trivially holds for a = c both sides being equal to 'a' by absorption. Since the diamond lattice is symmetric with respect to b1, b2, b3 the only situations with respect to the condition of the form a < c are b1 < 1 and 0 < b1.

Thus taking a = b1, and c = 1

a Ú (b Ù c) = b1 Ú (b Ù 1) = b1 Ú b

and (a Ú b) Ù c = (b1 Ú b) Ù 1 = b1 Ú b

whatever be b.

similarly for a = 0, c = b1.

a Ú (b Ù c) = 0 Ú (b Ù b1) = b Ù b1

while (a Ú b) Ù c = (0 Ú b) Ù b1 = b Ù b1 and so modularity holds.

Theorem :- A lattice (A, Ù, Ú) is modular if and only if it does not have a sublattice isomorphic with the pentagonal lattice.

Cont...