order theory unit 9 study guides

Key Concepts and Definitions

• Dimension theory studies the notion of dimension in partially ordered sets (posets) and lattices
• Ordered set $P$ has dimension $n$ if it can be embedded into the product of $n$ linear orders $L_1 \times L_2 \times \cdots \times L_n$
  • Embedding is an order-preserving map $f: P \to L_1 \times L_2 \times \cdots \times L_n$
• Order dimension $\text{dim}(P)$ is the smallest $n$ such that $P$ can be embedded into the product of $n$ linear orders
• Realizer of an ordered set $P$ is a collection of linear extensions whose intersection is $P$
  • Linear extension is a total order that extends the partial order on $P$
• Width $w(P)$ of an ordered set $P$ is the maximum size of an antichain in $P$
  • Antichain is a subset of $P$ in which no two elements are comparable
• Height $h(P)$ of an ordered set $P$ is the maximum size of a chain in $P$
  • Chain is a totally ordered subset of $P$

Historical Context and Development

• Dimension theory in ordered sets originated from the work of Dushnik and Miller in 1941
  • They introduced the concept of dimension for partially ordered sets
• Dilworth's theorem (1950) established a fundamental connection between the width and the minimum number of chains covering an ordered set
• Hiraguchi (1951) proved that the dimension of a finite ordered set is at most half its number of elements
• Kimble (1973) introduced the concept of interval dimension for ordered sets
  • Interval dimension considers the number of interval orders needed to realize an ordered set
• Trotter (1975) developed the theory of dimension for interval orders and semiorders
• Reuter (1985) introduced the concept of Boolean dimension for ordered sets
  • Boolean dimension considers embeddings into Boolean lattices
• Recent developments include the study of dimension for specific classes of ordered sets (posets of bounded height, planar posets)

Types of Dimensions in Ordered Sets

• Order dimension $\text{dim}(P)$ is the classical notion of dimension for an ordered set $P$
  • Minimum number of linear orders whose intersection is $P$
• Interval dimension $\text{idim}(P)$ considers the number of interval orders needed to realize $P$
  • Interval order is a poset where incomparability is transitive
• Boolean dimension $\text{bdim}(P)$ considers embeddings of $P$ into Boolean lattices
  • Boolean lattice is the poset of all subsets of a set ordered by inclusion
• Local dimension $\text{ldim}(P)$ is the maximum order dimension of the subposets induced by the elements incomparable to a given element
• Product dimension $\text{pdim}(P)$ is the minimum number of factor posets in a direct product representation of $P$
• Ferrers dimension $\text{fdim}(P)$ is the minimum number of Ferrers relations whose intersection is $P$
  • Ferrers relation is a binary relation satisfying certain properties
• Fractional dimension $\text{fdim}^*(P)$ is a relaxation of order dimension allowing fractional realizers

Properties and Characteristics

• Order dimension is monotone: if $P$ is a subposet of $Q$, then $\text{dim}(P) \leq \text{dim}(Q)$
• Order dimension is subadditive: $\text{dim}(P \cup Q) \leq \text{dim}(P) + \text{dim}(Q)$
• Order dimension is not additive: there exist posets $P$ and $Q$ such that $\text{dim}(P + Q) < \text{dim}(P) + \text{dim}(Q)$
  • $P + Q$ denotes the disjoint union of $P$ and $Q$
• Interval dimension is always less than or equal to the order dimension: $\text{idim}(P) \leq \text{dim}(P)$
• Boolean dimension is always greater than or equal to the order dimension: $\text{bdim}(P) \geq \text{dim}(P)$
• For a poset $P$ with width $w(P)$, the order dimension satisfies $\text{dim}(P) \leq w(P)$
• For a poset $P$ with height $h(P)$, the order dimension satisfies $\text{dim}(P) \leq h(P)$
• Standard examples of posets with high dimension include the Boolean lattice and the divisibility order on integers

Theorems and Proofs

• Dushnik-Miller theorem: For a poset $P$, $\text{dim}(P) \leq |P|/2$
  • Proof involves Dilworth's theorem and the decomposition of $P$ into chains
• Hiraguchi's inequality: For a poset $P$, $\text{dim}(P) \leq \lfloor |P|/2 \rfloor$
  • Strengthens the Dushnik-Miller theorem and is tight for Boolean lattices
• Kimble's theorem: For an interval order $P$, $\text{idim}(P) = \text{dim}(P)$
  • Establishes the equivalence of interval dimension and order dimension for interval orders
• Trotter's theorem: For a poset $P$, $\text{dim}(P) \leq w(P)$
  • Relates the order dimension to the width of the poset
• Felsner's theorem: For a planar poset $P$, $\text{dim}(P) \leq 4$
  • Shows that planar posets have bounded dimension
• Brightwell-Trotter theorem: For a poset $P$ with height $h(P)$, $\text{dim}(P) \leq 1 + \log_2(h(P))$
  • Provides an upper bound on the dimension in terms of the height
• Yannakakis's theorem: Determining whether a poset has dimension at most 3 is NP-complete
  • Establishes the computational complexity of the dimension problem

Applications in Mathematics

• Dimension theory is used in the study of graph drawing and graph coloring problems
  • Posets arising from graph coloring (chromatic number, fractional chromatic number) have connections to dimension
• Dimension theory has applications in the study of partial orders and lattices in combinatorics
  • Combinatorial problems on partially ordered sets (chains, antichains, linear extensions) relate to dimension
• Dimension theory is applied in the study of Boolean functions and circuit complexity
  • Posets associated with Boolean functions (minterm poset, maxterm poset) have connections to dimension
• Dimension theory is used in the analysis of algorithms and data structures
  • Partially ordered sets arising in algorithmic contexts (comparison-based algorithms, search trees) can be studied using dimension
• Dimension theory has applications in the field of social choice theory and decision making
  • Posets representing preferences and voting systems can be analyzed using dimension-theoretic tools
• Dimension theory is applied in the study of interval graphs and interval orders in graph theory
  • Interval dimension is a key parameter for characterizing interval graphs and orders

Connections to Other Areas of Order Theory

• Dimension theory is closely related to the study of linear extensions and realizers of posets
  • Linear extensions and realizers are fundamental objects in dimension theory
• Dimension theory has connections to the theory of interval orders and semiorders
  • Interval dimension and the dimension of interval orders are important research topics
• Dimension theory is related to the study of lattices and Boolean algebras
  • Boolean dimension considers embeddings into Boolean lattices
• Dimension theory has connections to the theory of well-quasi-ordering (WQO) and better-quasi-ordering (BQO)
  • Posets with bounded dimension often exhibit WQO or BQO properties
• Dimension theory is related to the study of order-preserving maps and embeddings between posets
  • Dimension can be characterized in terms of order-preserving maps and embeddings
• Dimension theory has connections to the theory of random partial orders and probabilistic methods in order theory
  • Probabilistic techniques are used to study the dimension of random posets

Advanced Topics and Current Research

• Study of dimension for specific classes of posets (planar posets, posets with bounded height, series-parallel posets)
  • Investigating the dimension of posets with special structural properties
• Complexity of computing the dimension and related parameters (interval dimension, Boolean dimension)
  • Developing efficient algorithms and studying the computational complexity of dimension problems
• Dimension theory for infinite posets and continuous posets
  • Extending dimension concepts to posets with infinite cardinality or continuous structures
• Connections between dimension and other poset parameters (width, height, jump number)
  • Exploring the relationships and trade-offs between dimension and other structural parameters
• Generalized dimension concepts (fractional dimension, local dimension, product dimension)
  • Developing and studying variations and generalizations of the classical dimension notion
• Dimension theory for ordered structures beyond posets (ordered graphs, ordered hypergraphs)
  • Extending dimension concepts to more general ordered structures and investigating their properties
• Applications of dimension theory in computer science, optimization, and combinatorial problems
  • Utilizing dimension-theoretic tools and insights in algorithmic and applied contexts