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An Introduction to Cwatsets and Graphs

The first cwatset was discovered in 1987 by Erich Friedman, then a student at Rose-Hulman Institute of Technology. Cwatsets were originally developed for statistical purposes, as a means of creating typical subsamples from sets that are not groups [1]. However, as the subject developed, it began to take on a much more algebraic flavor. Now, it is studied as much for its intrinsic value as an algebraic structure as for its statistical applications. Cwatsets have many parallels to group theory, and much of the study of the subject exploits group-theoretic results to yield more information about cwatsets. Interestingly enough, the subject has been developed almost exclusively by undergraduates, mostly students at Rose-Hulman and participants in programs at that school. Recently, relationships between cwatsets and graph theory have begun to arise. This paper is an introduction to the subject of cwatsets and explores some of these relationships with graph theory. To begin with an example, consider the subset F = {000, 110, 101} of the additive group Z2. (We write the elements of Z d 2 as binary strings for convenience.) Clearly, F is not a subgroup of Z2, since it is not closed under addition.