Here are five partially ordered sets (posets) along with their relations. You can use these to draw the Hasse diagrams and verify if they form a lattice:

1. Poset 1: Divisibility Relation

• Set: $P = \{1, 2, 4, 8\}$
• Relation: $a \leq b$ if $a$ divides $b$

2. Poset 2: Subset Relation

• Set: $P = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}$
• Relation: $A \leq B$ if $A \subseteq B$ (subset relation)

3. Poset 3: Power Set of 3-Element Set

• Set: $P = \{ \emptyset, \{x\}, \{y\}, \{z\}, \{x, y\}, \{x, z\}, \{y, z\}, \{x, y, z\} \}$
• Relation: $A \leq B$ if $A \subseteq B$

4. Poset 4: Divisibility on Powers of 3

• Set: $P = \{1, 3, 9, 27\}$
• Relation: $a \leq b$ if $a$ divides $b$

5. Poset 5: Custom Set with Binary Relation

• Set: $P = \{1, 2, 3, 6\}$
• Relation: $a \leq b$ if $a$ divides $b$

After drawing the Hasse diagram, check if each poset is a lattice by verifying:

1. Existence of least upper bound (join): For any two elements $a$ and $b$, their least upper bound exists.
2. Existence of greatest lower bound (meet): For any two elements $a$ and $b$, their greatest lower bound exists.

Would you like more details on the verification steps?

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Related Questions:

1. What is a lattice in the context of posets?
2. Can every poset be a lattice?
3. How do you determine the least upper bound (join) and greatest lower bound (meet)?
4. What is the significance of a Hasse diagram in visualizing posets?
5. Can you explain the difference between a bounded and unbounded lattice?

Tip: Always check if the poset has a least element (0) and greatest element (1) for completeness.