Lattices Practices Questions

Last Updated : 10 Jul, 2026

A lattice is a partially ordered set (POSET) in which every pair of elements has a unique least upper bound (join) and a unique greatest lower bound (meet). It is used to represent ordered relationships in mathematics, logic, and computer science.

Example:

In the set {1, 2, 3, 6} ordered by divisibility, the join of 2 and 3 is 6 (their least common multiple), and the meet is 1 (their greatest common divisor).

Question 1: Identify meet and join in a given set. Let L = {1, 2, 4, 8} be partially ordered by divisibility (a ≤ b if a divides b). Find:
1. 2 ∨ 4,
2. 2 ∧ 4

Solution:

  • Join (∨) = Least Common Multiple (LCM) in divisibility order

2 ∨ 4 = lcm(2, 4) = 4

  • Meet (∧) = Greatest Common Divisor (GCD) in divisibility order

2 ∧ 4 = gcd(2, 4) = 2

Question 2. Check if a given lattice is distributive. Consider L = {∅, {a}, {b}, {a, b}} ordered by set inclusion. Is L distributive?

Solution:

Meet = set intersection (∩)
Join = set union (∪)
For distributivity, check:

x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z) x and x ∪ (y ∩ z) = (x ∪ y) ∩ (x ∪ z)

For example, take x = {a}, y = {b}, z = {a, b}:
LHS = {a} ∩ ({b} ∪ {a,b}) = {a} ∩ {a,b} = {a}
RHS = ({a} ∩ {b}) ∪ ({a} ∩ {a,b}) = ∅ ∪ {a} = {a}

This holds for all cases → L is distributive.

Question 3: Modular law verification. Let L be the subspace lattice of {\displaystyle \mathbb {R} }^2 , where:

  • A = span{(1, 0)}
  • B = span{(0, 1)}
  • C = {\displaystyle \mathbb {R} }^2

Check if the modular law holds: A∨(B∧C) = (A∨B)∧C

Solution:

  • B ∧ C = B ∩ C = B
  • LHS = A ∨ B = span{(1,0),(0,1)} = R2
  • A∨B = {\displaystyle \mathbb {R} }^2

Modular law holds.

Question 4: Identify a non-modular lattice.

The pentagon lattice N5​ is given below (elements: 0, a, b, c, 1 with 0 < a < 1, 0 < b < c < 1, a and b incomparable). Show it is not modular.

Solution:

Take x = a, y = b, z = c where a ≤ c:

  • y ∧ z = b
  • LHS = a ∨ (b ∧ c) = a ∨ b = 1
  • x ∨ y = a ∨ b = 1
  • RHS = 1 ∧ c = c
    Since 1 ≠ c → violates modular law → not modular.

Question 5: Problem mixing meet, join, and complement

In the Boolean lattice P({1, 2, 3}) under set inclusion:

  • Meet = ∩
  • Join = ∪
  • Complement = relative to {1, 2, 3}

Find:

  1. Complement of {1, 2}
  2. {1, 2} ∨ {2, 3}
  3. {1, 2} ∧ {2, 3}

Solution:

  1. Complement = {3}
  2. Join = union → {1, 2, 3}
  3. Meet = intersection → {2}

Practice Questions

Q1. A lattice L = {1, 2, 4, 8, 16} is ordered by divisibility. Find the join (∨) and meet (∧) of the following pairs:

  • (4, 8)
  • (2, 16)
  • (8, 16)

Q2. Let L = {∅, {a}, {b}, {a, b}} be ordered by set inclusion.

  • Find the join (∨) and meet (∧) of {a} and {b}.
  • Find the complement of {a}.
  • Determine whether the lattice is bounded, complemented, and distributive.

Q3. A lattice satisfies the following properties:

  • Every pair of elements has a meet and a join.
  • It has a least element (0) and a greatest element (1).
  • Every element has a unique complement.

Identify the type(s) of lattice and justify your answer.

Q4. Determine whether each of the following statements is True or False. Give a reason for each answer.

  • Every finite lattice is complete.
  • Every complete lattice is bounded.
  • Every distributive lattice is modular.
  • Every complemented lattice is distributive.
  • Every bounded lattice is complete.

Q5. A lattice has elements {0, a, b, c, 1}, where 0 is the least element, 1 is the greatest element, a ≤ 1, b ≤ c ≤ 1, and a and b are incomparable.

Answer the following:

  • Is the lattice bounded?
  • Does it satisfy the modular law?
  • Can it be a distributive lattice? Explain your answer.
  • If it violates the modular law, what type of lattice does it represent?
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