Algebraic and Topological Combinatorics

Special Session at the Fall Eastern Section Meeting
of the American Mathematical Society

Williams College, Williamstown, MA
October 13-14, 2001

Kenneth P. Bogart
(Dartmouth College)

A geometric Classification of Trapezoid orders

Abstract [ps] [pdf]: We say that an order $P$ on a set $V$ is a {\em trapezoid order} if there are 2 lines (called baselines) parallel to the $x$-axis in the plane so that to each element $v$ in $V$ we may assign a trapezoid $T_v$ whose bases are on the baselines with $u\prec v$ if and only if their trapezoids do not intersect and every point in $T_u$, has a smaller $x$-coordinate than some point in $T_v$. Bases may have equal or zero length so that parallelograms and triangles are special cases of trapezoids. The assignment of trapezoids (parallelograms, triangles) to elements of $V$ is called a {\em trapezoid (parallelogram, triangle) representation} of $(V,P)$. An ordered set is called a {\em proper} or {\em unit} trapezoid (parallelogram, triangle) order if it has a representation in which no trapezoid (parallelogram, triangle) is a proper subset of any other trapezoid (parallelogram, triangle) or in which each trapezoid (parallelogram, triangle) has unit area. It turns out these properties yield 20 distinct classes of trapezoid orders. In particular the classes of proper trapezoid orders and unit trapezoid orders are distinct, which is the answer to the question that motivated this research. This talk will be a survey of why there are 20 classes and why they are distinct.

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last updated: August 21, 2001