  The GraphBLAS community has demonstrated the power of linear
algebra-leveraged graph algorithms, such as matrix-vector products for
breadth-first search (BFS) traversals. This paper investigates the algebraic
conditions needed for such computations when working with directed hypergraphs,
represented by incidence arrays with entries from an arbitrary value set with
binary addition and multiplication operations. Our results show the one-step
BFS traversal is equivalent to requiring specific algebraic properties of those
operations. Assuming identity elements 0, 1 for operations, we show that the
two operations must be zero-sum-free, zero-divisor-free, and 0 must be an
annihilator under multiplication. Additionally, associativity and commutativity
are shown to be necessary and sufficient for independence of the one-step BFS
computation from several arbitrary conventions. These results aid in
application and algorithm development by determining the efficacy of a value
set in computations.
