The image contains questions and exercises on posets (partially ordered sets), including concepts such as minimal and maximal elements, greatest lower bound (glb), least upper bound (lub), upper and lower bounds, and Hasse diagrams. Here’s a breakdown of the questions in English, along with brief descriptions:

Questions Summary

1. Fill in the blanks with the correct English terms for each concept related to posets.

   • For example:
     • Partially ordered set (poset) definition.
     • Linear order definition.
     • Greatest lower bound and least upper bound definitions.
     • Partial order requirements.

2. Identify specific elements within the given poset ({1, 2, 4}, {1}, {1, 2}, {1, 4}, {2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}) under the subset inclusion relation ⊆:

   • Minimal and maximal elements.
   • Greatest and least elements.
   • Determine upper and lower bounds for specific subsets.
   • Calculate the least upper bound and greatest lower bound for specified subsets, if they exist.

3. Determine if comparability exists between given elements in the poset.

4. Analyze a Hasse diagram to identify lattice structure and calculate supremum (least upper bound) properties for elements within the diagram.

Detailed Solution Process Outline

For each of the posed questions, here’s a potential approach:

1. Poset Terminology (1-5):

   • Use definitions to fill in terms like "linear order," "greatest lower bound," "least upper bound," and so on.

2. Element Identification in Poset (6-15):

   • Minimal and Maximal Elements (6-7): Look for elements that have no smaller or larger elements respectively, under the subset relation.
   • Greatest and Least Elements (8-9): If there exists an overall greatest or least element in the poset, find it.
   • Upper and Lower Bounds (10-13): Determine bounds for specific sets by examining subset relations.
   • Comparability (14-15): Decide if specific elements can be compared directly under the poset’s ordering relation.

3. Hasse Diagram Analysis (16-17):

   • Lattice Determination (16): Check if the diagram represents a lattice by seeing if every pair of elements has a unique least upper bound and greatest lower bound.
   • Supremum Calculation (17): Find the least upper bound in the context given by the Hasse diagram.

Would you like me to walk through each question with solutions, or focus on particular questions from this set?

Expanded Questions for Further Practice

1. How can we determine if a poset has a unique minimal or maximal element?
2. What distinguishes a lattice from a general poset?
3. How do you determine the least upper bound for a subset if it’s not explicitly given?
4. Can a poset have multiple greatest lower bounds? Why or why not?
5. What is the significance of a Hasse diagram in understanding the structure of a poset?

Tip

When working with posets, always start by identifying maximal and minimal elements, as they often provide insight into the overall structure and relationships between elements.