Which of the following best describes what makes a poset a lattice?
AEvery element has a unique predecessor and a unique successor
BEvery pair of elements has a least upper bound (join) and a greatest lower bound (meet)
CThe poset is finite and has both a top element and a bottom element
DThe partial order is total — every pair of elements is comparable
A lattice is a poset with the additional property that every pair of elements has a unique join (least upper bound) and a unique meet (greatest lower bound). Having a top and bottom element is insufficient — the condition must hold for every pair, not just globally. Total orders are lattices (join = max, meet = min), but not every lattice is a total order. Finiteness is irrelevant — the divisibility poset on all positive integers is an infinite lattice.
Question 2 Multiple Choice
In a Boolean lattice of all subsets of {1, 2, 3}, what is the meet of {1, 2} and {2, 3}?
A{1, 2, 3} — their union, the least set containing both
B{1, 3} — the elements not shared by both sets
C{2} — their intersection, the greatest set contained in both
D∅ — the bottom element of any Boolean lattice
In a Boolean lattice (power set ordered by inclusion), the meet is intersection and the join is union. The meet of {1,2} and {2,3} must be the greatest set that is a subset of both — that's their intersection {2}. Note that ∅ is also a subset of both, but {2} is larger, so {2} is the *greatest* lower bound. The join (union) would be {1,2,3}. This mirrors the Boolean algebra you know: AND = intersection = meet, OR = union = join.
Question 3 True / False
In a lattice, the join and meet of two elements must themselves be elements of the lattice — they cannot be external values.
TTrue
FFalse
Answer: True
This is part of the definition. The join a ∨ b is the least upper bound within the poset itself — it must be an element of the lattice that is ≥ both a and b, with no smaller element of the lattice also ≥ both. If you could appeal to external elements, every poset would trivially be a 'lattice.' The closure requirement is what makes lattice structure a genuine property of the poset.
Question 4 True / False
The join operation in a Boolean lattice corresponds to union of sets, and the meet corresponds to intersection.
TTrue
FFalse
Answer: True
For the power set of S ordered by inclusion, A ∨ B = A ∪ B (the smallest set containing both A and B) and A ∧ B = A ∩ B (the largest set contained in both A and B). This is also why Boolean algebra and set algebra share the same distributive laws: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) corresponds to A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Question 5 Short Answer
In the divisibility poset on positive integers, explain what join and meet represent and why this poset is a lattice.
Think about your answer, then reveal below.
Model answer: Join corresponds to LCM (least common multiple) and meet corresponds to GCD (greatest common divisor). For any two positive integers a and b, lcm(a,b) is the smallest integer divisible by both — the least upper bound under divisibility — and gcd(a,b) is the largest integer that divides both — the greatest lower bound. Since LCM and GCD always exist for positive integers, every pair has a join and meet, making this a lattice.
This example is valuable because it shows that you already knew lattice operations without the name: LCM and GCD are join and meet in the divisibility order. The lattice framework unifies these familiar operations with union/intersection in set theory and AND/OR in Boolean algebra — revealing a common algebraic structure across seemingly different contexts.