Upper Bounds

Reading the Symbol First

The symbol ≼ is just an ordering arrow. Read $$a ≼ b$$ as "a is below or equal to b" in the ordering.

It does not have to mean "smaller number". In a divisibility poset it means "a divides b".

So the same symbol carries a different meaning depending on which poset you are working in.

What an Upper Bound Is

Pick a subset S inside a poset $$(P, ≼)$$. An upper bound is any element x in P that sits above every member of S.

Formally: $$s ≼ x$$ for every s in S.

Notice the word every. One member of S failing the test disqualifies x immediately.

From Upper Bounds to the Least One

Usually several elements pass the upper bound test. Collect all of them.

The least upper bound (lub, or supremum) is the smallest element of that collected set.

So the work splits in two: first gather the whole set of upper bounds, then choose the smallest one inside it.

The Mirror Image Below

Lower bounds work the same way, flipped.

A lower bound is an element y in P with $$y ≼ s$$ for every s in S. Note the direction: y is now on the left of the symbol.

The greatest lower bound (glb, or infimum) is the largest element among all those lower bounds.

A Divisibility Example

Take $$P = \{1, 2, 3, 4, 6, 12\}$$ ordered by divisibility, and $$S = \{2, 3\}$$.

An upper bound must be divisible by both 2 and 3. That gives 6 and 12, so the least upper bound is 6.

A lower bound must divide both 2 and 3. Only 1 qualifies, so the greatest lower bound is 1.

Why 4 Fails the Test

The number 4 looks large, but size is not the ordering here.

Under divisibility, 3 does not divide 4, so $$3 ≼ 4$$ is false. One failure is enough to reject 4 as an upper bound.

This is the most common slip: reading ≼ as "less than or equal to" when the poset says otherwise.

Your Working Routine

Fix the meaning of ≼ for the poset in front of you.

Then run the two sweeps: collect every upper bound and take the smallest, collect every lower bound and take the largest.

Those two answers, the lub and the glb, are exactly what lattice questions ask you to produce for a pair of elements.

In a divisibility poset, does $$3 ≼ 4$$ mean that 3 divides 4?

• Yes
• No

Answer: No

No. In a divisibility poset, $$a ≼ b$$ means “a divides b.” Since 3 does not divide 4, $$3 ≼ 4$$ is false.

For S = {2, 3} in a divisibility poset, an _____ must be divisible by both elements, while a _____ must divide both elements.

Answer: For S = {2, 3} in a divisibility poset, an upper bound must be divisible by both elements, while a lower bound must divide both elements.

An upper bound lies above every member of S, so it must be divisible by both 2 and 3. A lower bound lies below every member, so it must divide both 2 and 3.

Let P = {1, 2, 3, 4, 6, 12} be ordered by divisibility, and let S = {2, 3}. What is the least upper bound of S?

• 1
• 4
• 6
• 12

Answer: 6

An upper bound must be divisible by both 2 and 3. The upper bounds are 6 and 12, and 6 is below 12 in the divisibility ordering. Therefore, the least upper bound is 6.