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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...

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3 votes

See Total Order Relation (TOSET) = Chain ( Poset + every element is comparable )

Lattice Non-comparable are also ok . 

So you can say every Toset is lattice but Every Lattice is not Toset.

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For a Poset to be a lattice:

i. Every pair of elements should have LUB (Least upper bound).

ii. Every pair of elements should have GLB (Greatest lower bound).

As we know, a Poset in which every element is related, is called Total ordered set. The Hasse diagram forms a linear chain. In such a chain we can easily find LUB and GLB for every pair of elements.

Hence Total ordered set is a lattice.

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