Let L be a set with a relation R which is transitive, anti-symmetric and reflexive and for any two elements a, b ∈ L, let the least upper bound lub(a,b) and the greatest lower bound glb(a,b) exist. Which of the following is true?
A. L is a poset
B. L is a Boolean algebra
C. L is a lattice
D. None of the above
GATE 1999 · Discrete Mathematics · Relations · medium
Answer: A and C (L is both a poset and a lattice)
L is a poset (option A): R is given as reflexive, antisymmetric, and transitive. This is exactly the definition of a partial order. Therefore (L, R) is a poset. Option A is TRUE.
L is a lattice (option C): We already know (L,R) is a poset. The problem states that for any two elements a, b in L, both lub(a,b) and glb(a,b) exist. This is precisely the definition of a lattice. Option C is TRUE.
L is not necessarily a Boolean algebra (option B): Boolean algebra requires the lattice to also be distributive (a meet (b join c) = (a meet b) join (a meet c)) and complemented (every element has a complement). Neither condition is given in the problem. So B is not guaranteed.