Zero Divisors Among Digraphs |
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Authors: | Richard Hammack Heather Smith |
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Institution: | 1. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, USA 2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA
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Abstract: | A digraph C is called a zero divisor if there exist non-isomorphic digraphs A and B for which ${A \times C \cong B \times C}$ , where the operation is the direct product. In other words, C being a zero divisor means that cancellation property ${A \times C \cong B \times C \Rightarrow A \cong B}$ fails. Lovász proved that C is a zero divisor if and only if it admits a homomorphism into a disjoint union of directed cycles of prime lengths.Thus any digraph C that is homomorphically equivalent to a directed cycle (or path) is a zero divisor. Given such a zero divisor C and an arbitrary digraph A, we present a method of computing all solutions X to the digraph equation ${A \times C \cong X \times C}$ . |
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