1.3 Hasse diagrams and their construction

Hasse diagrams visually represent partially ordered sets using vertices and edges. They show relationships between elements, making it easier to understand the structure and order of a poset. These diagrams are a key tool in lattice theory.

Hasse diagrams use directed acyclic graphs and apply transitive reduction to simplify the representation. They reveal hierarchical structures, symmetries, and relationships within posets, helping us analyze and compare different partially ordered sets more effectively.

Hasse Diagram Fundamentals

Visual Representation of Posets

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• Hasse diagrams provide a visual representation of partially ordered sets (posets)
• Consist of vertices connected by edges to illustrate the ordering relations between elements
• Each vertex represents an element of the poset
• Edges connect vertices to indicate the partial order relation
  • If $a \leq b$, then there is an upward path from vertex $a$ to vertex $b$

Graph Theory Concepts

• Hasse diagrams are a type of directed acyclic graph (DAG)
  • Directed edges have a specific orientation indicating the order relation
  • Acyclic means there are no cycles or loops in the graph
• Transitive reduction is applied to simplify the diagram
  • Removes redundant edges that can be inferred from transitivity
  • If $a \leq b$ and $b \leq c$, the edge from $a$ to $c$ is removed (transitivity implies $a \leq c$)
• Connected components refer to subgraphs of the Hasse diagram
  • Each connected component represents a subset of the poset that is not comparable to elements in other components
  • Useful for identifying independent or disjoint subsets within the poset

Hasse Diagram Structure

Levels and Hierarchical Organization

• Hasse diagrams often exhibit a hierarchical structure with distinct levels
• Elements at the same level are incomparable to each other
• Levels are determined by the length of the longest path from the minimal elements
  • Minimal elements have no incoming edges and form the bottom level
  • Maximal elements have no outgoing edges and form the top level
• The number of levels in a Hasse diagram depends on the poset's structure and partial order relations

Symmetry and Automorphisms

• Diagram automorphisms are symmetries or self-isomorphisms of the Hasse diagram
• An automorphism preserves the structure and order relations of the diagram
  • Permutes the vertices while maintaining the edges and their directions
• Automorphisms can reveal symmetries and structural properties of the poset
  • Rotational symmetry (e.g., a square lattice)
  • Reflectional symmetry (e.g., a diamond lattice)
• Studying automorphisms helps in understanding the underlying symmetries and regularities of the poset