The Zig-Zag Property and Exponential Cancellation of Ordered Sets |
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Authors: | McKenzie Ralph |
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Institution: | (1) Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, U.S.A. |
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Abstract: | This paper outlines a chapter in the theory of ordered sets that concerns decompositions of ordered sets using various natural
operations introduced by Garrett Birkhoff in 1940 – chiefly, the operations of sum (disjoint union), Cartesian product, and
exponentiation.
This area has seen two brief, four-year flowerings of research activity during the past sixty-four years. Ivan Rival, working
with several collaborators around 1978, obtained beautiful results on exponentiation which attracted other researchers to
this topic. Altogether, about a dozen people contributed a chain of significant new results between 1978–1982. The subject
then lay dormant until a major contribution by Jonathan D. Farley appeared in 1996. This pre-figured the solution, four years
later, of a central problem about exponentiation of finite ordered sets that Birkhoff had formulated in 1942.
The theory of operations on ordered sets is a beautiful chapter of mathematics. We concentrate on those portions that are
most relevant for the solution of the Birkhoff problem, offering (hopefully) accessible proofs of many results, including
some new results. We would like to see a new renaissance of activity in this area, which continues to offer many attractive
open problems.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | cancellation of exponents exponentiation posets |
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