TOSETs and Bounded Lattices

Question

• Which of the following are bounded lattices?

1.  (S, subset operation)
2. (Z, <=)

• Are all TOSETs bounded lattices?

Comments

thank you.

this cleared all my doubts.

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@hs_yadav A bounded lattice can be infinite.

Ex. ([0,1], <=) is infinite bounded lattice..

Am I right?

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1. (S, subset operation): yes this is a bounded lattice with phi as the least element and S as the greatest element. 
2. (Z, <=): no this is not a bounded lattice as this will be an infinite chain. 

For Bounded Lattice, Greatest and Least elements should exist. 

Only finite TOSET are bounded lattice. 

 

Answer 1

1. SUBSET OPERATION is bounded lattice

               UPPER BOUND = S  (set itself)

               LOWER BOUND = Phi (empty set)

 

LATTICE means only that EVRY TWO ELEMENTS SHOULD have glb and lub. WE WILL FIND THAT EACH SET has a lub and glb.

BUT FOR TOSET, we need FOR EVERY ELEMENT IN DOMAIN (a,b) , either (a,b) or (b,a) is in LATTICE.

2. UNBOUNDED TOSET

           LOWER BOUND => UNDEFINED  tending to[ -infinity] (every integer has an lesser element)

           UPPER BOUND => UNDEFINED tends to [ +infinity]  (every integer has an lesser element)

HERE EVERY ELEMENT IS COMPARABLE. SO TOSET