📊Order Theory Unit 9 – Dimension theory in ordered sets

Dimension theory in ordered sets explores how partially ordered sets can be embedded into products of linear orders. It studies various types of dimensions, including order dimension, interval dimension, and Boolean dimension, each providing insights into the structure of posets. Key concepts include realizers, width, and height of ordered sets. The field has connections to graph theory, combinatorics, and algorithm analysis. Recent research focuses on specific poset classes, computational complexity, and generalizations of dimension concepts.

Study Guides for Unit 9

9.1

Order dimension

7 min read

9.2

Dushnik-Miller dimension

4 min read

9.3

Realizers and linear extensions

7 min read

9.4

Dimension of special posets

6 min read

9.5

Fractional dimension

7 min read

9.6

Boolean dimension

5 min read

9.7

Computational aspects of dimension theory

9 min read

Key Concepts and Definitions

Dimension theory studies the notion of dimension in partially ordered sets (posets) and lattices
Ordered set $P$ $P$ has dimension $n$ $n$ if it can be embedded into the product of $n$ $n$ linear orders $L_1 \times L_2 \times \cdots \times L_n$ $L_{1} \times L_{2} \times \dots \times L_{n}$
- 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 $P$ $P$ is a collection of linear extensions whose intersection is $P$ $P$
- Linear extension is a total order that extends the partial order on $P$
Width $w(P)$ $w (P)$ of an ordered set $P$ $P$ is the maximum size of an antichain in $P$ $P$
- Antichain is a subset of $P$ in which no two elements are comparable
Height $h(P)$ $h (P)$ of an ordered set $P$ $P$ is the maximum size of a chain in $P$ $P$
- 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 $\text{dim}(P)$ $dim (P)$ is the classical notion of dimension for an ordered set $P$ $P$
- Minimum number of linear orders whose intersection is $P$
Interval dimension $\text{idim}(P)$ $idim (P)$ considers the number of interval orders needed to realize $P$ $P$
- Interval order is a poset where incomparability is transitive
Boolean dimension $\text{bdim}(P)$ $bdim (P)$ considers embeddings of $P$ $P$ 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 $\text{fdim}(P)$ $fdim (P)$ is the minimum number of Ferrers relations whose intersection is $P$ $P$
- 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 $P$ $P$ and $Q$ $Q$ such that $\text{dim}(P + Q) < \text{dim}(P) + \text{dim}(Q)$ $dim (P + Q) < dim (P) + dim (Q)$
- $P + Q$ 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 $P$ $P$ , $\text{dim}(P) \leq |P|/2$ $dim (P) \leq ∣ P ∣/2$
- Proof involves Dilworth's theorem and the decomposition of $P$ into chains
Hiraguchi's inequality: For a poset $P$ $P$ , $\text{dim}(P) \leq \lfloor |P|/2 \rfloor$ $dim (P) \leq ⌊ ∣ P ∣/2 ⌋$
- Strengthens the Dushnik-Miller theorem and is tight for Boolean lattices
Kimble's theorem: For an interval order $P$ $P$ , $\text{idim}(P) = \text{dim}(P)$ $idim (P) = dim (P)$
- Establishes the equivalence of interval dimension and order dimension for interval orders
Trotter's theorem: For a poset $P$ $P$ , $\text{dim}(P) \leq w(P)$ $dim (P) \leq w (P)$
- Relates the order dimension to the width of the poset
Felsner's theorem: For a planar poset $P$ $P$ , $\text{dim}(P) \leq 4$ $dim (P) \leq 4$
- Shows that planar posets have bounded dimension
Brightwell-Trotter theorem: For a poset $P$ $P$ with height $h(P)$ $h (P)$ , $\text{dim}(P) \leq 1 + \log_2(h(P))$ $dim (P) \leq 1 + lo g_{2} (h (P))$
- 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