Zero Divisors Among Digraphs

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 ${A \times C \cong X \times C}$ .