Total order Relation : Its also called "Linearly ordered set" or "chain". A poset(Partial order set) [ A ; R ] is called "total ordered set " if every pair of elements in Set A are comparable, in the sense ie;a R b or b R a , ∀ a , b ∈ A.
Lattice : A Poset is said to be lattice if both LUB(Least upper bound) and GLB(Greatest Lower Bound) exist.
for eg:
Set A contains divisors of 16 ={1,2,4,8,16}
if we make hasse diagram for this, then it would be linear chain.
in this diagram between every pair of elements , their exist GLB and LUB, so it is called lattice.
I think , there will be some kind of relationship(according to question) occurs between every pair of elements, that why elements will be comparable to each other..
Hence, I will say Every Total Order Relation is lattice but vice versa is not true...