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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2000-2001 MATHEMATICS COLLOQUIUM SERIES 04/03/01 Colloquium DR. PETER HILTON DEPARTMENT OF MATHEMATICS UNIVERSITY OF CENTRAL FLORIDA PROBLEMS CREATED BY NON-CANCELLATION Abstract: There are many interesting areas of mathematics, apart from the domain of numbers, in which we find binary operations, or even pairs of binary operations, where all the usual laws of arithmetic hold except that we do not have a cancellation law. For example, sets under union and intersection; pointed compact polyhedra under one-point union and the so-called smashed product; and finitely generated groups under direct product. In this talk we discuss the extent to which the usual procedures of arithmetic remain valid in the absence of cancellation laws. We will also show how to construct examples exhibiting the failure of cancellation.