Background Interval Restrictions Ground Set Extensions Intervals and Orders: What Comes After Interval Orders? Garth Isaak Department of Mathematics Lehigh University SIAM Discrete Mathematics Conference June 2006 Special session on the Mathematics of Kenneth P. Bogart Background Interval Restrictions Ground Set Extensions Background Interval Restrictions Ground Set Extensions Background Interval Restrictions Ground Set Extensions Background Interval Restrictions Ground Set Extensions Background Interval Restrictions Ground Set Extensions Background Interval Restrictions Ground Set Extensions Ken Bogart’s Ph.D. students 1. Issie Rabinovitch (1973) The Dimension theory of Semiorders and Interval Orders 9. Karen Stellpflug Mandych (1990) 2. T. Lockman Greenough (1976) Discrete representations of semiorders Representation and Enumeration of 10. Larry Langley (1993) Interval tolerance Interval Orders and Semiorders orders and dimension 3. Jean Gordon (1977) Applications of the 11. Sanjay Rajpal (1995) On paving Exterior Algebra to Graphs and Linear matroids representable over finite fields Codes 12. Stephen Ryan (1999) Trapezoid orders: 4. Donald Goldberg (1978) Generalized Geometric classification and dimension Weights for Linear Codes 13. Amy Myers (1999) Results in 5. Anita Solow (1978) The Square Class enumeration and topology of interval orders Invariant for Quadradic Forms 14. Laura Montague Hegerle (2000) Join 6. Mark Halsey (1984) Line Closure and congruence relations on geometric lattices the Steinitz Exchange Axiom: Hartmanis 15. Joshua Laison (2001) Tube Matroids representations of ordered sets 7. David Magagnosc (1987) Cuts and 16. David Neel (2002) Modular Chain Decompositions: Structure and contractibility in GF(q)-representable Algorithms matroids 8. Joseph Bonin (1989) Structural 17. Barry Balof (2003) Free Triangle Properties of Dowling Geometries and Orders: Results and Generalizations Lattices Background Interval Restrictions Ground Set Extensions Ken Bogart’s Ph.D. students 1. Issie Rabinovitch (1973) The Dimension theory of Semiorders and Interval Orders 9. Karen Stellpflug Mandych (1990) 2. T. Lockman Greenough (1976) Discrete representations of semiorders Representation and Enumeration of 10. Larry Langley (1993) Interval tolerance Interval Orders and Semiorders orders and dimension 3. Jean Gordon (1977) Applications of the 11. Sanjay Rajpal (1995) On paving Exterior Algebra to Graphs and Linear matroids representable over finite fields Codes 12. Stephen Ryan (1999) Trapezoid orders: 4. Donald Goldberg (1978) Generalized Geometric classification and dimension Weights for Linear Codes 13. Amy Myers (1999) Results in 5. Anita Solow (1978) The Square Class enumeration and topology of interval orders Invariant for Quadradic Forms 14. Laura Montague Hegerle (2000) Join 6. Mark Halsey (1984) Line Closure and congruence relations on geometric lattices the Steinitz Exchange Axiom: Hartmanis 15. Joshua Laison (2001) Tube Matroids representations of ordered sets 7. David Magagnosc (1987) Cuts and 16. David Neel (2002) Modular Chain Decompositions: Structure and contractibility in GF(q)-representable Algorithms matroids 8. Joseph Bonin (1989) Structural 17. Barry Balof (2003) Free Triangle Properties of Dowling Geometries and Orders: Results and Generalizations Lattices Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented by ‘less than’ on a set of intervals. Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented by ‘less than’ on a set of intervals. Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented by ‘less than’ on a set of intervals. Finite orders: can use intervals of real numbers Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented Some Variations: by ‘less than’ on a set Intervals in other of intervals. ordered sets Finite orders: can use Restrict Intervals intervals of real Relax ‘less than’ numbers Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented Some Variations: by ‘less than’ on a set Intervals in other of intervals. ordered sets Finite orders: can use Restrict Intervals intervals of real Relax ‘less than’ numbers Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented Some Variations: by ‘less than’ on a set Intervals in other of intervals. ordered sets Finite orders: can use Restrict Intervals intervals of real Relax ‘less than’ numbers Background Interval Restrictions Ground Set Extensions Interval Orders Interval Orders Can be represented Some Variations: by ‘less than’ on a set Intervals in other of intervals. ordered sets Finite orders: can use Restrict Intervals intervals of real Relax ‘less than’ numbers Background Interval Restrictions Ground Set Extensions Relax ‘less than’ I (Bounded) Tolerance Orders - a ≺ b if the interval for a is ‘below’ the interval for b and the overlap is ‘small enough’ I Equivalently - Tolerance Orders represented by ‘less than’ on a set of parallelograms with base on two lines I There are many variations on tolerance orders/graphs I Tolerance graphs are intersection graphs with tolerances. When there is a natural ordering they correspond to tolerance orders I Bogart wrote a half dozen papers on tolerance graphs/orders Background Interval Restrictions Ground Set Extensions Relax ‘less than’ I (Bounded) Tolerance Orders - a ≺ b if the interval for a is ‘below’ the interval for b and the overlap is ‘small enough’ I Equivalently - Tolerance Orders represented by ‘less than’ on a set of parallelograms with base on two lines I There are many variations on tolerance orders/graphs I Tolerance graphs are intersection graphs with tolerances. When there is a natural ordering they correspond to tolerance orders I Bogart wrote a half dozen papers on tolerance graphs/orders Background Interval Restrictions Ground Set Extensions Relax ‘less than’ I (Bounded) Tolerance Orders - a ≺ b if the interval for a is ‘below’ the interval for b and the overlap is ‘small enough’ I Equivalently - Tolerance Orders represented by ‘less than’ on a set of parallelograms with base on two lines I There are many variations on tolerance orders/graphs I Tolerance graphs are intersection graphs with tolerances. When there is a natural ordering they correspond to tolerance orders I Bogart wrote a half dozen papers on tolerance graphs/orders Background Interval Restrictions Ground Set Extensions Relax ‘less than’ I (Bounded) Tolerance Orders - a ≺ b if the interval for a is ‘below’ the interval for b and the overlap is ‘small enough’ I Equivalently - Tolerance Orders represented by ‘less than’ on a set of parallelograms with base on two lines I There are many variations on tolerance orders/graphs I Tolerance graphs are intersection graphs with tolerances. When there is a natural ordering they correspond to tolerance orders I Bogart wrote a half dozen papers on tolerance graphs/orders Background Interval Restrictions Ground Set Extensions Relax ‘less than’ I (Bounded) Tolerance Orders - a ≺ b if the interval for a is ‘below’ the interval for b and the overlap is ‘small enough’ I Equivalently - Tolerance Orders represented by ‘less than’ on a set of parallelograms with base on two lines I There are many variations on tolerance orders/graphs I Tolerance graphs are intersection graphs with tolerances. When there is a natural ordering they correspond to tolerance orders I Bogart wrote a half dozen papers on tolerance graphs/orders Background Interval Restrictions Ground Set Extensions Replace parallelogram with trapezoids to get interval dimension 2 orders (intersection of 2 interval orders) N Bogart suggested extending bounded tolerance orders (parallelogram orders) to other geometric figures and ‘more lines’. Results by Bogart and his students Ryan, Laison, Balof ≺ These problems can be viewed as investigating when a given order is a suborder of an (infinite) ordered set described by geometric objects. Background Interval Restrictions Ground Set Extensions Replace parallelogram with trapezoids to get interval dimension 2 orders (intersection of 2 interval orders) N Bogart suggested extending bounded tolerance orders (parallelogram orders) to other geometric figures and ‘more lines’. Results by Bogart and his students Ryan, Laison, Balof ≺ These problems can be viewed as investigating when a given order is a suborder of an (infinite) ordered set described by geometric objects. Background Interval Restrictions Ground Set Extensions Replace parallelogram with trapezoids to get interval dimension 2 orders (intersection of 2 interval orders) N Bogart suggested extending bounded tolerance orders (parallelogram orders) to other geometric figures and ‘more lines’. Results by Bogart and his students Ryan, Laison, Balof ≺ These problems can be viewed as investigating when a given order is a suborder of an (infinite) ordered set described by geometric objects. Background Interval Restrictions Ground Set Extensions Replace parallelogram with trapezoids to get interval dimension 2 orders (intersection of 2 interval orders) N Bogart suggested extending bounded tolerance orders (parallelogram orders) to other geometric figures and ‘more lines’. Results by Bogart and his students Ryan, Laison, Balof ≺ These problems can be viewed as investigating when a given order is a suborder of an (infinite) ordered set described by