Question

Homomorphic image of modular lattice is modular

Answer 1

Answer:

Let L be modular lattice, M be lattice and f:L→M be a homomorphism. I want to show f(L) is a modular lattice..

We already know that homomorphic image of a lattice is a lattice. So we only want to show that if f(a)≤f(b) then f(a)∨(f(x)∧f(b))=(f(a)∨f(x))∧f(b) for a,b,x∈L

Since L is modular a≤b implies a∨(x∧b)=(a∨x)∧b

My problem is:

I must begin with the assumption f(a)≤f(b) then show f(a)∨(f(x)∧f(b))=(f(a)∨f(x))∧f(b) for which I need to use a∨(x∧b)=(a∨x)∧b

But f(a)≤f(b) need not necessarily imply a≤b since it is only homomorphism and not isomorphism.

hope i help u......